DEPARTMENT OF CIVIL ENGINEERING BANGLADESH …

EVALUATION OF SEISMIC PERFORMANCE OF RC FRAME

STRUCTURES BY PUSHOVER AND TIME HISTORY ANALYSES

MD. NASIM KHAN

MASTER OF SCIENCE IN CIVIL ENGINEERING (STRUCTURAL)

DEPARTMENT OF CIVIL ENGINEERING

BANGLADESH UNIVERSITY OF ENGINEERING AND

TECHNOLOGY, DHAKA, BANGLADESH

JUNE, 2013

ii

The thesis titled “EVALUATION OF SEISMIC PERFORMANCE OF RC FRAME

STRUCTURES BY PUSHOVER AND TIME HISTORY ANALYSES” submitted by

Md. Nasim Khan, Roll No: 100604336P, Session: October/2006; has been accepted as

satisfactory in partial fulfillment of the requirement for the degree of Master of

Science in Civil Engineering (Structural) on 29 June, 2013.

BOARD OF EXAMINERS

___________________________ Dr. Tahsin Reza Hossain Chairman Professor (Supervisor) Department of Civil Engineering BUET, Dhaka-1000. ____________________________ Dr. Md. Mujibur Rahman Professor and Head Member (Ex-Officio) Department of Civil Engineering BUET, Dhaka-1000. ____________________________ Dr. Mahbuba Begum Associate Professor Member Department of Civil Engineering BUET, Dhaka-1000. ____________________________ Dr. Alamgir Habib Professor (Retd), Dept. of Civil Engg, BUET Member (External) Professor, MIST Urban Heights, Apartment # 2A, House # 124 Road # 9/A, Dhanmondi, Dhaka.

iii

Declaration

It is hereby declared that except for the contents where specific reference have been

made to the work of others, the studies contained in this thesis is the result of

investigation carried out by the author. No part of this thesis has been submitted to

any other University or other educational establishment for a Degree, Diploma or

other qualification (except for publication).

Signature of the Candidate

_____________________________

(Md Nasim Khan)

iv

Dedication

To

My

Mother

v

Contents Declaration iii

Contents v

List of Tables ix

List of Figures x

Acknowledgment xvi

Abstract xvii

Chapter 1 Introduction 1-5

1.1 Background and present state of the problem 1

1.2 Objectives with specific aims and possible outcomes 3

1.3 Methodology 3

1.4 Layout of thesis 4

Chapter 2 Literature review 6-45

2.1 Introduction 6

2.2 Provision for earthquake load in BNBC 6

2.2.1 Equivalent static force method 7

2.2.2 Dynamic response method 8

2.2.2.1 Response spectrum method 8

2.3 Selection of lateral force method BNBC, 2006 9

2.4 Pushover Analysis 9

2.4.1 Methods to perform simplified nonlinear analysis 10

2.4.2 Capacity 11

2.4.3 Demand (displacement) 11

2.4.4 Performance 12

2.5.5 Reduced demand spectra 12

2.4.6 Development of elastic site response spectra 14

2.4.7 Seismic zone 14

2.4.8 Seismic source type: 15

2.4.9 Near source factor 16

2.4.10 Seismic coefficients 17

2.4.11 Development of elastic site response spectra 19

2.4.12 Establishing demand spectra 19

2.4.13 Capacity spectrum method 22

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2.4.14 Displacement coefficient method 30

2.4.15 Seismic performance evaluation 30

2.4.16 Nonlinear static procedure for capacity evaluation of structures 31

2.4.17 Structural performance levels and ranges 32

2.4.17.1 Immediate occupancy structural performance level (S-1) 33

2.4.17.2 Damage control structural performance range (S-2) 33

2.4.17.3 Life safety structural performance level (S-3) 33

2.4.17.4 Limited safety structural performance range (S-4) 34

2.4.17.5 Collapse prevention structural performance level (S-5) 34

2.4.17.6 Target building performance levels 35

2.4.18 Response limit 35

2.4.18.1 Global building acceptability limits 35

2.4.18.2 Element and component acceptability limit 37

2.4.19 Acceptability limit 39

2.5 Nonlinear time history analysis 40

2.5.1 Nonlinear dynamic analysis for earthquake ground motions 41

2.5.2 Linearly elastic and inelastic systems 41

2.5.3 Equation of motion for seismic vibration 42

2.5.4 Solution by incremental time-step integration 43

2.5.5 The average acceleration method 44

2.9 Conclusion 45

Chapter 3 Pushover Analysis 46-74

3.1 Introduction 46

3.2 Details of pushover analysis in SAP2000 and ETABS 47

3.3 Description of case study frame for validation 51

3.3.1 Two story RC frame 51

3.3.2 Five story RC frame 53

3.3.3 Twelve story RC frame 54

3.4 Analysis and results of validation of pushover curve for 2D frame 56

3.5 Comparison of pushover curve for SAP2000 and SeismoStruct 58

3.5.1 Description of SeismoStruct 58

3.5.2 Description of case study frames 60

3.5.3 Analysis and results 60

3.5.3.1 Two storied 2D frame 61

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3.5.3.2 Five storied 2D frame 62

3.6 Different loading pattern for pushover curve 63

3.6.1 Two storied 2D frame 63

3.6.2 Five storied 2D frame 64

3.6.3 Twelve storied 2D frame 64

3.7 Performance evaluation of structure 65

3.7.1 Two story 2D frame 66

3.7.1.1 Local level performance 68

3.7.1.2 Global level performance 69

3.7.2 Five story 2D frame 70



3.7 Conclusion 74

Chapter 4 Time History Analysis 75-108

4.1 Introduction 75

4.2 Time history analysis using selected earthquakes 76

4.2.1 El Centro 1940 Earthquake 77

4.2.2 Kobe Earthquake 78

4.3 Validation of linear time history analysis for SDOF system 78

4.3.1 Description of SDOF Model 78

4.3.2 Analysis and Result 79

4.4 Nonlinear Time History Analysis 82

4.4.1 Validation of nonlinear time history analysis for SDOF system with

SeismoStruct 83

4.4.1.1 Description of model 83

4.4.1.2 Structural geometry and properties 86

4.4.1.3 Modeling and loading 86

4.4.1.4 Analysis type 87

4.4.1.5 Comparison of analysis results 89

4.5 Nonlinear time history analysis for 2D frame using SAP2000 92

4.5.1 Analysis Technique 93

4.5.2 Analysis and results 94

4.6 Comparison of Linear and Nonlinear Time History Analysis 96

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4.7 Comparison of capacity curves of pushover and nonlinear time

history analysis 99

4.7.1 Description of case studies model 100

4.7.2 Analysis and Result 100

4.8 Performance Evaluation of the Structure using time history 103

4.8.1 Local level performance 103

4.8.2 Global level performance 105

4.8 Conclusion 108

Chapter 5 Performance based analysis of RC frame building 109-143

5.1 Introduction 109

5.2 Performance requirements 109

5.3 Description of 6-story reinforced concrete frame structure building 110

5.3.1 Performance evaluation of a structure design as per BNBC 112



5.4 Performance of 6-story RC narrow building 122




5.5 Performance of 12-story RC frame building 131




5.6 Performance evaluation of structure for gravity loads only 141

5.7 Conclusion 143

Chapter 6 Conclusions and future recommendations 144-146

6.1 Introduction 144

6.2 Findings of the study 145

6.3 Recommendation of future study 146

References 148-153

Appendix- A 1-4

Appendix- B 5-10

Appendix- C 11-18

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List of Tables _____________________________________________________________________ Table 2.1 Minimum allowable SRA and SRV values (Adopted from ATC 40) 13

Table 2.2 Seismic zone factor Z 14

Table 2.3 Seismic source type as per ATC-40 16

Table 2.4 Seismic source factor 16

Table 2.5 Seismic coefficient CA 17

Table 2.6 Seismic coefficient CV 18

Table 2.7 Soil profile types (ATC-40) 18

Table 2.8 Calculation of CA 19

Table 2.9 Calculation of CV 20

Table 2.10 Deformation Limits (ATC-40) 36

Table 2.11 Examples of possible deformation controlled and force controlled

actions (FEMA-356) 38

Table 3.1 Cross section and loading properties of 2-story RC frame 52

Table 3.2 Dynamic properties of 2-Story RC frame 53


Table 3.4 Dynamic properties of 5-story RC frame 54


Table 3.6 Dynamic properties of 12-story RC frame 56



Table 4.1 Comparison time history analysis with ETABS of SDOF

system and published result from Chopra 79

Table 5.1 Structural dimension of building 110



Table 5.4 Calculation of Reduction factor 113

Table 5.5 Capacity spectrum of the six story building (Procedure A and B) 117

Table 5.6 Number of hinges formed in the 6-storied frame structure

building in x and y-direction 118

Table 5.7 Structural dimension of 6-story (aspect ratio) building 122


Table 5.9 Capacity spectrum of the six story building (Procedure A and B) 128

x



Table 5.11 Structural dimension of 12-story building 132


Table 5.13 Capacity spectrum of the 12-story building (Procedure A and B) 138



Table 5.15 Number of hinges for different level of earthquakes 141

List of figures

_____________________________________________________________________

Fig. 2.1 Normalized response spectra for 5% damping ratio

(Adopted from BNBC) 8

Fig. 2.2 Reduced response spectrum 12

Fig. 2.3 Seismic Zoning map of Bangladesh 15

Fig. 2.4 Typical capacity curve 20

Fig. 2.5 Code specified response spectrum in Spectral acceleration vs. Period 23

Fig. 2.6 Response spectrum in ADRS format 23

Fig. 2.7 A typical capacity curve 24

Fig. 2.8 Capacity spectrum 26

Fig. 2.9 Typical capacity spectrum 27

Fig. 2.10 Determination of performance point (Adopted from ATC 40) 28

Fig. 2.11 Component force versus deformation Curves (FEMA-356) 37

Fig. 2.12 Force-deformation action and acceptance criteria (ATC-40) 39

Fig. 2.13 Nonlinear force-displacement relationship 44

Fig. 3.1 Concrete moment and P-M-M hinge property 48

Fig. 3.2 Concrete Shear hinge property 48

Fig. 3.3 An example to illustrate the option of save positive increments only 50

Fig. 3.4 Two story RC frame 52

Fig. 3.5 Five story RC frame 53

Fig. 3.6 Twelve story RC frame 55

Fig. 3.7 Pushover curves of the 2-story 2D frame 57

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Fig. 3.8 Pushover curves of the 5-story 2D frame 57

Fig. 3.9 Pushover curves of the 12-Story 2D frame 57

Fig. 3.10 Pushover curves of the 2-story 2D frame for different loading

pattern using SAP2000 and SeismoStruct 61


pattern using SAP2000 and SeismoStruct 62


pattern using SAP2000 64





Fig. 3.15 Capacity spectrum of the 2-story 2D frame (Procedure A) 66

Fig. 3.16 Capacity spectrum of the 2-story frame for SE (Procedure B) 67

Fig. 3.17 Capacity spectrum of the 2-story frame for DE (Procedure B) 67

Fig. 3.18 Capacity spectrum of the 2-story frame for ME (Procedure B) 68

Fig. 3.19 Deformation of the 2-story frame at SE level 68

Fig. 3.20 Deformation of the 2-story frame at DE level 69

Fig. 3.21 Deformation of the 2-story frame at ME level 69

Fig. 4.22 Story drift ratio at performance point of 2-Story 2D frame for

different earthquake level 70

Fig. 3.23 Capacity spectrum of the 5-story 2D frame (Procedure A) 70

Fig. 3.24 Capacity spectrum of the 5-story frame for SE (Procedure B) 71

Fig. 3.25 Capacity spectrum of the 5-story frame for DE (Procedure B) 71

Fig. 3.26 Capacity spectrum of the 5-story frame for ME (Procedure B) 72

Fig. 3.27 Deformation of the 5-story frame at SE level 72

Fig. 3.28 Deformation of the 5-story frame at DE level 73

Fig. 3.29 Deformation of the 5-story frame at ME level 73

Fig. 3.30 Story drift ratio at performance point of 5-Story 2D frame for

different earthquake level 74

Fig. 4.1 Ground acceleration of El Centro, 1940 earthquake record (N-S),

adopted from Chopra [5] 77

Fig. 4.2 Ground acceleration of Kobe Earthquake 78

Fig.4.3 SDOF Model 79

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Fig. 4.4(a) Time history analysis of SDOF system to El Centro ground motion

for Tn= 0.5 sec, ξ=2% 80

Fig. 4.4(b) Time history analysis of SDOF system to El Centro ground motion

for Tn= 1 sec, ξ=2% 80

Fig. 4.4(c) Time history analysis of SDOF system to El Centro ground motion


Fig. 4.5(a) Time history analysis of SDOF system to El Centro ground motion


Fig. 4.5(b) Time history analysis of SDOF system to El Centro ground motion


Fig. 4.5(c) Time history analysis of SDOF system to El Centro ground motion


Fig. 4.6 Input Ground motion (EQ1) 84






Fig. 4.12 Pier cross section and bridge pier specimen configuration 86

Fig. 4.13 FE model of the bridge column 87

Fig. 4.14 Analytical results at top displacement with time (EQ1 to EQ6) 88

Fig. 4.15 Analytical results at base shear with time (EQ1 to EQ6) 88

Fig. 4.16 Analytical results at moment with time (EQ1 to EQ6) 89

Fig. 4.17 Experiment vs. Analytical results at top displacement time (EQ1) 89



Fig. 4.20 Experiment vs. Analytical results at base shear time (EQ1) 91



Fig. 4.23 Time History Analysis for 5-Story 2D Frame (El Centro earthquake) 95

Fig. 4.24 Time History Analysis for 5-Story 2D Frame (Kobe earthquake) 95

Fig. 4.25 Comparison of linear and nonlinear time history analysis for 2-Story

frame (El Centro Earthquake) 96

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frame (Kobe Earthquake) 97









Fig. 4.31 Capacity Curve for 2-Story Frame (El Centro earthquake) 102



Fig. 4.34 Deformation of the 2-Story 2D frame for ME (El Centro earthquake) 104

Fig. 4.35 Deformation of the 2-Story 2D frame for ME (Kobe earthquake) 104

Fig. 4.36 Deformation of the 5-Story 2D frame for ME (El Centro earthquake) 105

Fig. 4.37 Deformation of the 5-Story 2D frame for ME (Kobe earthquake) 105

Fig. 4.38 Maximum story drift at performance point of 2-Story 2D frame for

different earthquake level (El Centro earthquake) 106


different earthquake level (Kobe earthquake) 106


different earthquake level (El Centro earthquake) 107


different earthquake level (Kobe earthquake) 107

Fig. 5.1 Layout of the 6-story building 110

Fig. 5.2 Capacity spectrum of the six-story frame structure building in x-

direction (Procedure A) 113

Fig. 5.3 Capacity spectrum of the six-story frame structure building in y-



direction (SE) (Procedure B) 114


direction (DE) (Procedure B) 115

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direction (ME) (Procedure B) 115







Fig. 5.10 Deformation of the building at performance point in x-direction for SE 119

Fig. 5.11 Deformation of the building at performance point in x-direction for DE 119

Fig. 5.12 Deformation of the building at performance point in x-direction for ME 119

Fig. 5.13 Deformation of the building at performance point in y-direction for SE 120

Fig. 5.14 Deformation of the building at performance point in y-direction DE 120

Fig. 5.15 Maximum story drift ratio at performance point for different

earthquake level in X-direction 121


earthquake level in Y-direction 121

Fig. 5.17 Layout of the 6-story building (aspect ratio) 122

















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earthquake level in x-direction 130


earthquake level in y-direction 130

Fig. 5.28 Layout of the 12-story building 131

Fig. 5.29 Elevation of 12-story building 132

Fig. 5.30 Capacity spectrum of the 12-story frame structure building in x-direction

(Procedure A) 134

Fig. 5.31 Capacity spectrum of the 12-story frame structure building in y-direction

(Procedure A) 134


(SE) (Procedure B) 135


(DE) (Procedure B) 135


(ME) (Procedure B) 136


(SE) (Procedure B) 136


(DE) (Procedure B) 137


(ME) (Procedure B) 137









xvi

Acknowledgement

The author wishes to convey his profound gratitude to Almighty Allah for His

graciousness, unlimited kindness and blessings, and for allowing him to complete the

thesis.

The author wishes to express his sincere appreciation and gratitude to his supervisor,

Dr. Tahsin Reza Hossain, Professor, Department of Civil Engineering, BUET, Dhaka,

for his continuous guidance, invaluable suggestions and continued encouragement

throughout the progress of the research work.

The author is also grateful to Dr. Khan Mohammed Amanat, Professor, Department of

Civil Engineering, BUET, Dhaka, for his guidance and inspiration at the initial stage

of the study.

A very special debt of deep gratitude is offered to his mother and brothers for their

continuous encouragement and cooperation during this study. Last but not the least;

the author is deeply indebted to his beloved wife for supporting him all the time with

love and inspiration.

xvii

Abstract

Bangladesh is situated in a seismically active region. Part of the country extended

from Sylhet to Chittagong is in the high seismic zone whereas Dhaka lies in the

moderate seismic zone. Major metropolitan cities of our country are under serious

threat because of inadequacies in design and construction of structures. A lot of

multistoried buildings have already been built in order to fulfill the ever increasing

demand of urban population. Bangladesh National Building Code (BNBC) proposes

equivalent static load method to design the buildings; however a seismic event will

result in damaged structure, performance of which cannot be evaluated from a static

analysis. Pushover analysis, a comparatively simplified nonlinear method involving

certain approximations and simplifications, is capable of predicting the damage extent

due to a seismic event. ATC-40 and FEMA-356 propose the pushover analysis

method in detail. In this thesis, an attempt has been made to demonstrate the validity

and efficiency of pushover analysis method of ATC-40 as incorporated in ETABS,

SAP2000 and SeismoStruct software. Performance of building frames designed as per

BNBC has also been evaluated against targeted performance levels for serviceability,

design and maximum earthquakes. Nonlinear time history analysis, although

complicated and time consuming, is a more rigorous method of modeling seismic

response of a structure. Both linear and nonlinear time history analyses have been

validated against published results using SAP2000 and SeismoStruct. The work also

studied the effectiveness of pushover analysis in comparison to more rigorous

nonlinear time history analysis with particular emphasis on the load pattern employed

in pushover analysis. Different load patterns have been used in pushover analysis and

uniform load pattern has been found to give better capacity curves that compare well

with nonlinear time history analysis. Performances have also been evaluated for 2D

frames designed according to BNBC using nonlinear time history. Well-known EL

Centro and Kobe earthquakes have suitably been scaled as per required Z value for

use in the analysis. A comparison of linear and nonlinear time history analyses have

been carried out to demonstrate the damage extent caused during an earthquake.

Similar pushover analyses carried out on 3-D six and twelve story building designed

according to BNBC show that they easily satisfy the ATC-40 local and global seismic

requirements. A building which is designed only for gravity load failed to satisfy the

requirements for serviceability and maximum earthquakes.

1

Chapter 1

Introduction 1.1 Background and present state of the problem Bangladesh is situated in a seismically active region. Part of the country extended from Sylhet

to Chittagong is in the high seismic zone whereas Dhaka is in the moderate seismic zone.

Major metropolitan cities of our country are under serious threat because of inadequacies in

design and construction of structures. Rapid urbanization creates a great demand on human

shelter especially in big cities like Dhaka, Chittagong etc. As a result, a lot of multistoried

buildings are already built in order to fulfill the demand.

There was no written building code in Bangladesh until 1993. In 1993, Bangladesh National

Building Code (BNBC) was published by Housing and Building Research Institute (HBRI)

which is commonly known as BNBC [1]. The seismic design provisions of BNBC were based

on the UBC [2]. Since then, BNBC has widely been used by engineers. BNBC has different

provisions for earthquake load calculation and analysis procedure. For the regular structures,

the Code defines a simple method to represent earthquake induced inertia forces by Equivalent

Static Force for static analysis. For very tall structure or for irregular structure, the Code

provisions require more rigorous analysis, namely, 1) Response Spectrum Analysis and 2)

Time History Analysis. All these methods detailed in BNBC are force based methods. As in

many other codes, the level of forces prescribed by BNBC for a structure is rather arbitrarily set

and aimed at damage control performance objectives i.e. no damage under small earthquake

and no collapse under extreme earthquake. The code approach is to design seismic load

resisting system on the basis of a pseudo-seismic load obtained by dividing the actual load by

response modification factor, R. The R value is specified by the code for each structural system

without explicitly defining the level of element (i.e. beam, column, connection etc.) ductility

required for each system. The code implicitly assumes that the enhanced ductile detailing

would result in seismic energy dissipation and hence a reduced demand would result.

Especially the wide availability of computer technology has made a more realistic simulation of

structural behavior possible under seismic loading. The focus of seismic design in current

building codes is one of life safety level. Economic losses due to recent earthquakes are

estimated to be billions of Taka and the numbers will be higher if the indirect losses are

included. This fact lets code committees and decision makers think beyond life safety, which is

2

essential in design, to alleviate economic losses. This trend creates an increased interest in

performance-based design for structure. As a result of which a number of methods are proposed

by ATC-40 [3], FEMA-356 [4] for a more rational analysis of the structures under seismic

loading. These are simplified nonlinear static analysis capable of simulating the degradation of

structure. Nonlinear time history analysis is a more rigorous and correct method but requires

huge computing and post-processing effort [5]. One of the main advantages of performance-

based designs is its ability to show the performance situation of the structure and its

components under different load intensities. The performance situation means that the damage

level, if any, can be assessed and a judgment can be made as to which degree this structure can

continue to service.

Extensive research studies have already been carried out regarding the performance of concrete

structures. Comartin et.al [6] gives a practical overview of the ATC-40 method in seismic

evaluation and retrofit of concrete buildings. Fajfar and Fischinger [7] proposed as a simple

nonlinear procedure for seismic damage analysis of reinforced concrete buildings. The method

uses response spectrum approach and nonlinear static analysis. Krawinkler and Seneviratna [8]

conducted a detailed study that discusses the advantages, disadvantages and the applicability of

pushover analysis by considering various aspects of the procedure. Ýnel et. al. [9] conducted a

study to evaluate the accuracy of various lateral load patterns used in current pushover analysis

procedures. First mode, inverted triangular, rectangular, adaptive lateral load patterns and

multimode pushover analysis were studied. Chopra and Goel [10] developed an improved

pushover analysis procedure named as Modal Pushover Analysis (MPA) which is based on

structural dynamics theory. Sasaki, et al. [11] proposed Multi-Mode Pushover (MMP)

procedure to identify failure mechanisms due to higher modes. The procedure uses independent

load patterns based on higher modes besides the one based on fundamental mode. Kalkan and

Chopra [12] presented a modal-pushover-based scaling (MPS) procedure to scale ground

motions for use in nonlinear RHA of buildings. Oguz [13] carried out pushover analyses along

with nonlinear time history analyses to evaluate the accuracy of various lateral load patterns.

Many important works have also been carried out in Bangladesh regarding pushover analysis

with focus on seismic deficiencies and remedies [14], stiffness irregularity in vertical direction

[15], soft story vulnerability [16]. Nonlinear time history analysis for RC frames with brick

masonry infill has also been carried out [17]. Hossen and Anam [18], Abdullah et.al. [19]

studied nonlinear moment-curvature (M-φ) relationship of beam and column sections and used

3

them in a nonlinear dynamic analysis to investigate the performance of different structural

systems. However, the accuracy of pushover analysis in comparison to more rigorous nonlinear

time history analysis has not yet been studied. Also, it is important to ascertain whether

building frames designed by BNBC [1] are capable of fulfilling the targeted performance

levels.

1.2. Objectives with specific aims and possible outcomes

The specific objectives of the proposed study are as follows:

1) To introduce a simplified nonlinear analysis method, i.e. pushover method, for generation of

capacity and demand curves of a structure and determine its performance.

2) To validate the simplified nonlinear static analysis i.e. pushover analysis as incorporated in

different softwares with published numerical results.

3) To perform a comparative study of pushover analysis and time history analysis with a view

to study the effectiveness of pushover analysis.

4) To determine the performance levels for the design of different concrete frames according to

BNBC [1] seismic provisions.

5) To validate linear and nonlinear time history analyses for SDOF system using different

softwares with published numerical and experimental results.

This work aims to demonstrate the validity and efficiency of pushover analysis method of

ATC-40 as incorporated in ETABS [20] and SAP2000 [21]. The work would also study the

effectiveness of pushover analysis in comparison to more rigorous nonlinear time history

analysis with particular emphasis on the load pattern employed in pushover analysis. Finally

performance of building frames designed as per BNBC [1] will be evaluated against targeted

performance levels for serviceability, design and maximum earthquakes.

1.3. Methodology

In order to achieve the selected objectives, the research work is initiated by studying seismic

provisions of current codes of practices, available literatures on nonlinear static and dynamic

analyses methods. The adequacy of the simplified nonlinear analysis i.e. pushover analysis as

4

given in ETABS/ SAP2000/SeismoStruct will be validated with published numerical and

experimental results. The analysis is capable of progressive damage of elements by inserting

appropriate hinges as the structure is laterally pushed through. Geometric non-linearity (P-∆

effect) is also included in the analysis. The resulting capacity curve (Base shear vs.

displacement) is superimposed on earthquake demand curve in the same domain. The point of

intersection of theses curves will represent structure’s performance level. However, the

capacity curve depends on the load-pattern assumed in the simplified nonlinear analysis. By

comparing with more rigorous nonlinear time history analyses, accuracy of pushover analysis is

studied. Particular emphasis is given on the load pattern employed in pushover analysis.

Typical five to twelve storied buildings representative of low to high-rise buildings which are

very common in Bangladesh is designed as per the force level detailed in BNBC and pushover

analysis is carried out to evaluate the different performance levels under serviceability, design

and maximum earthquakes.

The general purpose finite element program e.g. SAP2000, ETABS and SeismoStruct are the

primary tool for modeling the structures and studying its behavior in terms of capacity and

performance.

1.4. LAYOUT OF THESIS

The general background and present state of the problem, objectives with specific aims,

possible outcomes and scope of the thesis and methodology of the work are discussed in

Chapter 1, which give the basic idea of the study and its methodology. Chapter 2 describes

pushover analysis, Time history analysis for different types of earthquake loading. This

chapter also presents M-Φ relation for pushover analysis. In Chapter 3, nonlinear static

analysis procedures in terms of capacity and demand curves have been studied. Modeling

parameters and acceptance criteria for nonlinear hinges are also presented in this chapter. A

site response spectrum with 5% damping for Dhaka is also developed in this chapter. In this

chapter, pushover analysis results have been validated by comparing with published

numerical results. Also, procedure of performance based analysis of concrete structure using

software SAP2000/ETABS have been described. Chapter 4 mainly deals with nonlinear time

history analysis, comparison with pushover analysis and validation of linear and nonlinear

time history analysis with published results. In Chapter 5, performance based analysis of 3D

RC frame building, designed according to BNBC, has been carried out to determine its

5

performance under three levels of earthquakes. A building not designed for seismic loads

have also been studied to determine its seismic performance. Chapter 6 presents the finding

of the research work, conclusion along with the recommendations for future study. The

appendices are also included to present the ATC-40 provisions.

6

Chapter 2

Literature review 2.1 Introduction

Majority of buildings in our country are still designed for gravity loads only with nominal

non seismic detailing provisions. The knowledge and application of seismic detailing is

very limited among the structural designers of Bangladesh. This is quite unexpected,

particularly since the Bangladesh National Building Code BNBC [1] contains a chapter on

detailing of reinforced concrete structures (PART 6, Chapter 8). A recently published

earthquake resistant design manual by Bangladesh Earthquake Society is a significant

addition to this type of literature in Bangladesh, which had been practiced worldwide in

earthquake prone areas [e.g., seismic design code provisions of American Concrete

Institute ACI 318 (1999), Euro code 8 (2002) and Indian Standards IS 1893 (2002) and IS

13920 (2002)].

Many important works have also been carried out in Bangladesh regarding pushover

analysis with focus on seismic deficiencies and remedies [14], stiffness irregularity in

vertical direction [15], soft story vulnerability [16]. Nonlinear time history analysis for RC

frames with brick masonry infill has also been carried out [17]. Hossen and Anam [18],

Abdullah et.al.[19] studied nonlinear moment-curvature (M-φ) relationship of beam and

column sections and used them in a nonlinear dynamic analysis to investigate the

performance of different structural systems. However, the accuracy of pushover analysis in

compare to more rigorous nonlinear time history analysis has not yet been studied. Also, it

is important to ascertain whether building frames designed by BNBC [1] are capable of

fulfilling the targeted performance levels.

2.2 Provisions for earthquake load in BNBC

Bangladesh National Building Code BNBC [1] has different provisions for calculation of

earthquake load and analysis procedures for structures subjected to earthquake. Two

methods are available in BNBC for determination of seismic lateral forces on primary

framing systems.

7

2.2.1 Equivalent static force method

In this method the dynamic earthquake effect is represented by an equivalent static load at

different levels in proportion to mass at that level. In this process BNBC divided the

country into three region of different possible earthquake ground response (0.075g, 0.015g,

and 0.25g).

The total design base shear for a seismic zone is given by,

WR

ZICV = …………………………………………. (a)

Where,

Z= seismic zone co-efficient

I= Structural importance co-efficient

C= Numerical co-efficient =

T = Time period =

Where,

Ct = 0.083 for moment resisting frame

= 0.073 for reinforced concrete frame and eccentric braced still frame

= 0.049 for all other structural analysis

hn = Height in meter above base level n

S = Site co-efficient

R= Response modification co-efficient

W= Total seismic dead load

32

25.1

T

S

( )43

nt hC

8

Lateral force calculated from the above equation known as base shear V, shall be distributed along the height of the structure in accordance with the following equation

………………………………………………………………………….. (b)

Where, Fi = Lateral force applied at story level I and

Ft =Concentrated lateral force considered at the top of the building in addition to the force Fn. 2.2.2 Dynamic response method

2.2.2.1 Response spectrum method

BNBC recommends that response spectrum to be used in dynamic analysis shall be either

of the following:

a) Site specific design spectra: A site specific design spectra shall be developed base on

the geologic, tectonic, seismologic, and soil characteristics associated with the specific site.

b) Normalized response spectra: In absence of a site-specific response spectrum, the

normalized response spectra shall be used.

The normalized response spectrum curves provided in the code are prepared for three

different soil types and 5% of critical damping Fig. 2.1 below

Fig 2.1 Normalized response spectra for 5% damping ratio (Adopted from BNBC)

9

2.3 Selection of lateral force method in BNBC

Seismic lateral forces on primary framing systems shall be determined by using either the

equivalent static force method or the dynamic response method complying with the

restrictions given below:

a) The equivalent static force method may be used for the following structures:

i) All Structures, regular or irregular, in Seismic Zone 1 and Structure Importance

Category iv in Seismic Zone 2, except case b (iv) below.

ii) Regular structures less than 75 meters in height with lateral force resistance

provided by structural systems listed in BNBC except case b (iv) below.

iii) Irregular structures not more than 20 meters in height.

b) The dynamic response method may be used for all classes of structure, but shall be used

for the structures of the following types.

i) Structures 75 meters or more in height except as permitted by case a (i) above.

ii) Structures having a stiffness, weight or geometric vertical irregularity of type I,

II, III as defined in BNBC.

iii) Structures over 20 meters in height in Seismic Zone 3 not having the same

structural system throughout their height.

iv) Structures regular or irregular, located on soil type S4 as defined in BNBC.

2.4 Pushover analysis

An elastic analysis gives a good indication of the elastic capacity of structures, but it cannot

predict failure mechanisms and account for redistribution of forces during progressive

yielding for an earthquake excitation. Inelastic analyses procedures help demonstrate how

buildings really work by identifying modes of failure and potential for progressive collapse.

The use of inelastic procedures for design and evaluation is attempts to help engineers

better understand how structures will behave when subjected to major earthquakes, where it

is assumed that the elastic capacity of the structure will be exceeded. Application of this

resolves some of uncertainties associated with code and elastic procedures.

Various analysis methods are available, both linear and nonlinear for evaluation of concrete

building. The best basic inelastic method is nonlinear time history analysis method. This

10

method is too complicated and considered impractical for general use. The central focus of

this thesis is to introduce the simplified nonlinear procedure for the generation of the

“pushover” or capacity curve of a structure. Pushover analysis is a simplified static

nonlinear analysis method which use capacity curve and reduced response spectrum to

estimate maximum displacement of a building under a given level of earthquake.

2.4.1 Methods to perform simplified nonlinear analysis

As a structure responds to earthquake ground motion, it experiences lateral displacements

and, in turn, deformations of its individual elements. At low levels or response, the element

deformations will be within their elastic (linear) range and no damage will occur. At higher

levels of response, element deformations will exceed their linear elastic capacities and the

some of structural components will experience damage. In order to provide reliable seismic

performance, a structure must have a complete lateral force resisting system, capable of

limiting earthquake induced lateral displacements to levels at which the damage sustained

by the structural elements will be within acceptable levels for the intended performance

objective. The basic factors that affect the lateral force resisting system’s ability to do this

include the mass, stiffness, damping and configuration; the deformation capacity of the

building elements; and the strength and character of the ground motion it must resist.

The nonlinear pushover analysis requires development of the capacity curve .The capacity

curve is derived from an approximate nonlinear, incremental static analysis for the

structure. This capacity curve is simply a plot of the total lateral seismic shear demand,

“V,” on the structure, at various increments of loading, against the lateral deflection of the

building at the roof level, under that applied lateral force. The slope of a straight line drawn

from the origin of the plot for this curve to a point on the curve at any lateral displacement,

“d,” represents the secant or “effective” stiffness of the structure when pushed laterally to

that displacement.

Two key elements of a performance based design procedure are demand and capacity.

Demand is the representation of the earthquake ground motion. Capacity is the

representation of the structure’s ability to resist the seismic lateral force. The performance

11

is dependent on the manner that the capacity is able to handle the demand. In other words,

the structure must have the capacity to resist the demands of the earthquake such that the

performance of the structure is compatible with the objectives of the design.

Simplified nonlinear analysis procedures using pushover methods, such as the capacity

spectrum method and the displacement coefficient method, require determination of three

primary elements: capacity, demand (displacement) and performance. Each of these

elements is briefly discussed below.

2.4.2 Capacity

The overall capacity of a structure depends on the strength and deformation capacities of

the individual components of the structure. In order to determine capacities beyond the

elastic limits, some form of nonlinear analysis, such as the pushover procedure, is required.

This procedure uses a series of sequential elastic limits, some form of nonlinear analysis,

superimposed to approximate a force displacement capacity diagram of the overall

structure. The mathematical model of the structure is modified to account for reduced

resistance of yielding components. A lateral force distribution is again applied until

additional components yield. This process is continued until the structure becomes unstable

or until a predetermined limit is reached. The capacity curve approximates how structures

behave after exceeding their elastic limit.

2.4.3 Demand (displacement)

Ground motions during an earthquake produce complex horizontal displacement pattern in

structures that may vary with time. Tracking this motion at every time-step to determine

structural design requirements is judged impractical. Traditional linear analysis methods

use lateral forces to represent a design condition. For nonlinear methods it is easier and

more direct to use a set of lateral displacements as a design condition. For a given structure

and ground motion, the displacement demand is the estimate of the maximum expected

response of the building during the ground motion.

12

Spectral acceleration, Sa

Spec

tral d

eman

d, S

d

2.4.4 Performance

Once a capacity curve and demand displacement is defined, a performance check can be

done. A performance check verifies that structural and nonstructural components are not

damaged beyond the acceptable limits of the performance objective for the forces and

displacement imposed by the displacement demand.

2.4.5 Reduced demand spectra

The capacity of a particular building and the demand imposed upon it by a given

earthquake motion are not independent. One source of this mutual dependence is evident

from the capacity curve itself. As the demand increases the structure eventually yields and,

it’s stiffness decreases, it’s period lengthens. Conversion of the capacity curve to spectral

ordinates (ADRS) makes this concept easier to visualize. Since the seismic acceleration

depends on period, demand also changes as the structure yields. Another source of mutual

dependence between capacity and demand is effective damping. As a building yield in

response to seismic demand it dissipates energy with hysteretic damping. Building that

have large, stable hysteresis loops during cyclic yielding dissipate more energy then those

with pinched loops caused by degradation of strength and stiffness.

Fig. 2.2 Reduced response spectrum

2.5 C

Cv/T SRA (2.5Ca)

SRv (Cv /T)

13

Since the energy that is dissipated need not be stored in the structure, the effective damping

diminishes displacement demand. The reduced displacement demand is shown in Fig.2.2.

The equation for the reduced factor SRA and SRV are given by:

12.2)ln(681.021.3 eff

ASRβ−

= ≥Value in Table 2.1

65.1)ln(41.031.2 eff

VSRβ−

= ≥ Value in Table 2.1

Table 2.1: Minimum allowable SRA and SRV values1 (Adopted from ATC-40)

Structural Behavior Type SRA SRV

Type A2 0.33 0.50

Type B 0.44 0.56

Type C. 0.56 0.67

1. Values for SRA and SRV shall not be less than those shown in this Table

2. Type A, B and C is taken as defined in ATC 40 (1996)

Severity of earthquakes as classified in ATC-40, 1996 is defined below.

A. The serviceability earthquake (SE)

The serviceability earthquake (SE) is defined probabilistically as the level of ground

shaking that has a 50 percent chance of being exceeded in 50-year period. This level of

earthquake ground shaking is typically about 0.5 times of the level of ground shaking of the

design earthquake. The SE has a mean return period of approximately 75 years. Damage in

the nonstructural elements is expected during serviceability earthquake.

14

B. The design earthquake (DE)

The design earthquake (DE) is defined probabilistically as the level ground shaking that has

a 10 percent chance of being exceeded in a 50-year period. The DE represents an

infrequent level of ground shaking that can occur during the life of the building. The DE

has a mean return period of approximately 500 years. Minor repairable damage in the

primary lateral load carrying system is expected during design earthquake. For secondary

elements, the damage may be such that they require replacement.

C. The maximum earthquake (ME)

The maximum earthquake (ME) is defined deterministically as the maximum level of

earthquake ground shaking which may ever be accepted at the building site within the

known geologic frame work. In probabilistic terms, the ME has a return period of about

1,000 years. During maximum earthquake, buildings will be damaged beyond repairable

limit but will not collapse.

2.4.6 Development of elastic site response spectra

Elastic response spectra for a site are based on estimate of Seismic Coefficient, CA which

represents the effective peak acceleration (EPA) of the ground and CV which represents 5

percent damped response of a 1-second system. These coefficients for a particular zone are

dependent on the seismicity of the area, the proximity of the site to active seismic sources,

and site soil profile characteristics.

2.4.7 Seismic zone

Bangladesh is divided into three seismic zones as per BNBC. The table below shows the

values of zone coefficients of Bangladesh.

Table 2.2 Seismic zone factor Z

Zone 1 2 3

Z 0.075 0.15 0.25

15

Fig 2.3 Seismic Zoning map of Bangladesh

2.4.8 Seismic source type:

As per ATC-40 (1996), three types of seismic source may be defined as shown in Table 2.3

16

Table 2.3 Seismic source type as per ATC-40, 1996

Seismic source definition

Seismic

Source Type

Seismic Source Description Maximum moment

magnitude, M

Slip rate,

SR(mm/yr)

A Faults that are capable to produce large

magnitude events and which have a high

rate of seismic activity

M≥7.0 SR≥5

B All faults other than types A and C Not applicable Not applicable

C Faults that are not capable to producing

large magnitude events and which have

a high rate of seismic activity

M<6.5 SR<2

2.4.9 Near source factor

Currently data pertaining to the active faults close to Dhaka city is not available. It is not

possible to estimate the seismic source distance from a specific site being considered in this

thesis. But it may be safely assumed that all the sources are located at distance more than

15 km and the Table 2.4 (ATC-40, 1996) may be used to consider the Near-Source effects

for the present study.

Table 2.4 Seismic source factor

Seismic

source

type

Closed distance to known seismic source

≤2km 5 km 10 km ≥15 km

NA NV NA NV NA NV NA NV

A 1.5 2.0 1.2 1.6 1.0 1.2 1.0 1.0

B 1.3 1.6 1.0 1.2 1.0 1.0 1.0 1.0

C 1.0 1.0 1.0 1.0 1.0 1.0 1.0 1.0

1. The near source factor may be used on the linear interpolation of values for distance

other than those shown in the table.

17

2. The closest distance of the seismic source shall be taken as the minimum distance

between the site and the area described by the vertical projecting of source on the surface

(i.e., surface projection of fault plane). The surface projecting need not include portions of

the source a depths of 10km or greater. The largest value of the near-source factor

considering all sources shall be used for design.

2.4.10 Seismic coefficients

For each earthquake hazard level, the structure is assigned a seismic coefficient CA in

accordance Table 2.5 (ATC-40, 1996) and a seismic coefficient CV in accordance with

Table 2.6 (ATC-40, 1996). Seismic coefficient CA represents the effective peak

acceleration (EPA) of the ground. A factor of about 2.5 times CA represents the average

value of peak response of a 5 percent-damped short-period system in the acceleration

domain. The seismic coefficient CV represents 5 percent-damped response of a 1-second

system. CV divided by period (T) defines acceleration response in the velocity domain.

These coefficients are dependent on soil profile type and the product of earthquake zoning

coefficient-Z, severity of earthquake-E and near source factor-N (ZEN). The soil profile

types are classified in Table 2.7.

Table 2.5 Seismic coefficient CA

Soil profile type Shaking intensity, ZEN1,2

0.075 0.15 0.20 0.30

SB 0.08 0.15 0.20 0.30

SC 0.09 0.18 0.24 0.33

SD 0.12 0.22 0.28 0.36

SE 0.19 0.30 0.34 0.36

SF Site-specific geo-technical investigation required to determine CA

1. The value of E used to determine the product, ZEN, should be taken to be equal to

0.5 for the serviceability earthquake, 1.0 for the design earthquake, and 1.25 for the

maximum earthquake.

18

2. Seismic coefficient CA should be determined by linear interpolation for values of

the product ZEN other than those shown in the table.

Table 2.6 Seismic coefficient CV (Adopted from ATC-40)

Soil profile type Shaking intensity, ZEN1,2

0.075 0.15 0.20 0.30

SB 0.08 0.15 0.20 0.30

SC 0.13 0.25 0.32 0.45

SD 0.18 0.32 0.40 0.54

SE 0.26 0.50 0.64 0.84

SF Site-specific geo-technical investigation required to determine CV

1. The value of E used to determine the product, ZEN, should be taken to be equal to

0.5 for the serviceability earthquake, 1.0 for the design earthquake, and 1.25 for the

maximum earthquake.

2. Seismic coefficient CV should be determined by linear interpolation for values of

the product ZEN other than those shown in the table.

Table 2.7 Soil Profile Types (ATC-40)

Average soil properties for top 100 ft of soil profile

Soil Profile

Type

Soil profile

name/Generic

description

Share Wave

velocity,

Vs(ft/sec)

Standard penetration

Test, N or NCH for

cohesion less soil

layers(blow/ft)

Undrained shear

strength, Su(psf)

S1A Hard rock VS > 5,000 Not Applicable

SB Rock 2,500 < VS ≤

5,000

Not Applicable

SC Very dense soil

and rock

1,200 < VS ≤

2,500

N > 50 SU > 2,000

SD Stiff soil profile 600 <VS ≤ 1,200 15 ≤ N ≤ 50 1,000 ≤ SU ≤2,000

S2E Soft soil profile VS <600 N < 50 SU <1,000

SF Soil requiring site-specific evaluation

Soil profile SA is not applicable to site in Dhaka.

19

Soil profile type SE also include any soil profile with more than 10 feet or soft clay defined

as a soil with PI > 20, WMC ≥ 40 and Su < 500 psf..

2.4.11 Development of elastic site response spectra

Elastic response spectra for a site are based on estimate of seismic coefficient, CA which

represents the effective peak acceleration (EPA) of the ground and CV which represents 5

percent damped response of a 1-second system. These coefficients for a particular zone are

dependent on the seismicity of the area, the proximity of the site to active seismic sources,

and site soil profile characteristics.

2.4.12 Establishing demand spectra

For the purpose of subsequent analysis to be made in this thesis, it is necessary to establish

an earthquake demand spectra against which building performance will be evaluated. The

following controlling parameters are considered:

Location of the site : Dhaka City

Soil profile at the site : Soil type SE as per Table 4.6, soft soil with shear wave velocity

VS<600 ft/sec, N < 50 and Su < 100 psf

Earthquake source type : A – considering the events similar to the great Indian

Earthquake in Assam in 12 June, 1897

Near Source Factor : > 15km

Table 2.8: Calculation of CA

Seismic zone factor, Z 0.15 As per BNBC

Earthquake hazard level, E 1 Design Earthquake

Near source factor, N 1 >15km, Table 2.4

Shaking intensity, ZEN 0.15 >15km, Table 2.5

For soil type SE, CA 0.3

20

Table 2.9: Calculation of CV

Seismic zone factor, Z 0.15 As per BNBC

Earthquake hazard level, E 1 Design earthquake

Near source factor, N 1 >15km, Table 2.4

Shaking intensity, ZEN 0.15 From Table 2.6

For Soil Type SE, CV 0.5

A typical capacity curve of a hypothetical structure is shown in Fig. 2.4.

Fig. 2.4 Typical capacity curve

In Fig. 2.4, the discrete points indicated by the symbol ‘•’ represent the occurrence of

important events in the lateral response history of the structure. Such an event may be the

initiation of yield in a particulars structural element or a particular type of damage. Each

point is determined by a different analysis sequence. Then, by evaluating the cumulative

effects of damage sustained at each of the individual events, and the overall behavior of the

structure’s increasing lateral displacements, it is possible to determine and indicate on the

capacity curve those total structural lateral displacements that represent limits on the

various structural performance levels, as has been done in Fig. 2.4.

The process of defining lateral deformation points on the capacity curve at which specific

structural performance levels may be said to have occurred requires the exercise of

21

considerable judgment on the part of the engineer. For each of the several structural

performance levels and global performance levels defined in this chapter. It defines global

system response limits as well as acceptance criteria for the individual structural elements.

These acceptance criteria generally consist of limiting values of element deformation

parameters, such as the plastic chord rotation of a beam or shear angle of a wall. These

limiting values have been selected as reasonable approximate estimates of the average

deformations at which certain types of element behavior such as cracking, yielding, or

crushing, may be expected to occur. As the incremental static nonlinear analyses are

performed, the engineer must monitor the cumulative deformations of all important

structural elements and evaluate them against the acceptance criteria set before.

The point on the capacity curve at which the first element exceeds the permissible

deformation level for a structural performance level does not necessarily represent that the

structure as a whole reaches that structural performance level. Most structures contain

many elements and have considerable redundancy. Consequently, the onset of unacceptable

damage to a small percentage of these elements may not represent an unacceptable

condition with regard to the overall performance of the structure. When determining the

points along the capacity curve for the structure at which the various structural performance

levels may said to be reached, the engineer must view the performance of the structure as

whole and consider the importance of damage predicted for the various elements on the

overall behavior of the structure.

The methodology described by ATC-40, incorporates the concept of “Primary” and

“Secondary” elements to assist the engineer in making these judgments. Primary elements

are those that are required as part of the lateral force resisting system for the structure. All

other elements are designated as secondary elements. For a given performance level,

secondary elements are generally permitted to sustain more damage than primary elements

since degradation of secondary elements does not have a significant effect on the lateral

load resisting capability of the structure. If in the development of the capacity curve it is

determined that a few elements fail to meet the acceptance criteria for a given performance

level at an increment of lateral loading and displacement, the engineer has the ability to

designate these “nonconforming” elements as secondary, enabling the use of more liberal

22

acceptance criteria for these few elements. Care is exercised not to designate an excessive

number of elements that are effective in resisting lateral forces as secondary.

2.4.13 Capacity spectrum method

The capacity spectrum method, a nonlinear static procedure, provides a graphical

representation of the global force-displacement capacity curve of the structure (i.e.

pushover curve) and compares it to the response spectra representations of the earthquake

demands. This method is a very useful tool in the evaluation and retrofit design of both

existing concrete structures. The graphical representation provides a clear picture of how a

structure responds to earthquake ground motion, and, as illustrated below, it provides an

immediate and clear picture of how various retrofit strategies, such as adding stiffness or

strength, will affect the structure response to earthquake demands.

The capacity spectrum curve for the structure is obtained by transforming the capacity

curve from lateral force (V) vs. lateral displacement (d) coordinates to spectral acceleration

(Sa) vs. spectral displacement (Sd) coordinates using the modal shape vectors, participation

factors and modal masses obtained from a modal analysis of the structure. In order to

compare the Structure’s capacity to the earthquake demand, it is required to plot the

response spectrum and the capacity spectrum on the same plot. The conventional response

spectrum plotted in spectral acceleration vs. period coordinate has to be changed in to

spectral acceleration vs. spectral displacement coordinate. This form of response spectrum

is known as acceleration displacement response spectrum (ADRS).

Capacity spectrum method requires plotting the capacity curve in spectral acceleration and

spectral displacement domain. This representation of spectral quantities is knows as

Acceleration displacement response spectra in brief ADRS, which was introduced by

Mahaney et al.,(1993). Spectral quantities like spectral acceleration, spectral displacement

and spectral velocity is related to each other to a specific structural period T. Building code

usually provide response spectrum in spectral acceleration vs. period format which is the

conventional format.

23

Each point on the curve defined in the Fig. 2.5 is related to spectral displacement by

mathematical relation, 2

241 TSS ad π

=. Converting with this relation response spectrum in

ADRS format may be obtained.

Fig. 2.5 Code specified response spectrum in Spectral acceleration vs. Period.

Fig. 2.6 Response spectrum in ADRS format

24

Any line from the origin of the ADRS format represent a constant period Ti which is

related to spectral acceleration and spectral displacement by the mathematical relation,

a

d

SS

T π2=

Capacity Spectrum Capacity spectrum is a simple representation of capacity curve in

ADRS domain. A capacity curve is the representation of Base Shear (V) to roof

displacement (Xd). In order to develop the capacity spectrum from a capacity curve it is

necessary to do a point by point conversion to first mode spectral coordinates.

Fig. 2.7 A typical capacity curve

Any point corresponding values of base shear, Vi and roof deflection, ∆i may be converted to the corresponding point of spectral acceleration, Sai and spectral displacement, Sdi on the capacity spectrum using relation,

1

/α

WVS i

ai = and

Roof

Roofdi PF

S,11 Φ×

∆=

t

25

⎥⎥⎥⎥

⎦

⎤

⎢⎢⎢⎢

⎣

⎡

=

∑

∑

gw

gwPF

ii

N

N

ii

/)(

/)(

1,2

1

11,

1

φ

φ

Modal mass coefficient for the first mode, α1 is calculated using equation,

⎥⎦

⎤⎢⎣

⎡⎥⎦

⎤⎢⎣

⎡

⎥⎦

⎤⎢⎣

⎡

=

∑∑

∑N

ii

N

i

N

ii

gwgw

gw

11,

2

1

2

11,

1

/)(/

/)(

φ

φα

Where:

PF1 = modal participation factor for the first natural mode.

α1 = modal mass coefficient for the first natural mode

Φ1, roof = roof level amplitude of the first mode.

wi/g = mass assigned to level i

Φi1 = amplitude of mode 1 at level i

N = level N, the level which is the uppermost in the main portion of the

structure

V = base shear

W = building dead weight plus likely live loads

∆roof = roof displacement

Sa = spectral acceleration

Sd = spectral displacement

26

Fig 2.8 Capacity spectrum

Fig. 2.8 shows a typical capacity spectrum converted from capacity curve of Fig. 2.7 of a

hypothetical structure. It is seen in the capacity spectrum that up to some displacement

corresponding to point A, the period is constant T1. That is the structure is behaving

elastically. As the structure deflects more to point B, it goes to inelastic deformation and its

period lengthens to T2.

When the capacity curve is plotted in Sa vs. Sd coordinates, radial lines drawn from the

origin of the plot through the curve at various spectral displacements have a slope (ω),

where, ω is the radial frequency of the effective (or secant) first-mode response of the

structure if pushed by an earthquake to that spectral displacement.

Using the relationship T=2π/ω, it is possible to calculate, for each of these radial lines, the

effective period of the structure if it is pushed to a given spectral displacements.

Fig. 2.9 is a capacity spectrum plot obtained from the capacity curve of a hypothetical

structure shown in Fig. 2.4 and plotted with the effective modal periods shown.

27

Fig. 2.9 Typical capacity spectrum

The particular structure represented by this plot would have an elastic period of

approximately ½ second. As it is pushed progressively further by stronger ground motion,

this period lengthens. The building represented in Figs. 2.4 and 2.9 would experience

collapse before having its stiffness degraded enough to produce an effective period of 2

seconds.



from the capacity curve itself. As the demand increases, the structure eventually yields and,

as its stiffness decreases, its period lengthens. Conversion of the capacity curve to spectral

ordinates (ADRS) makes this concept easier to visualize. Since the seismic accelerations

depend on period, demand also changes as the structure yields. Another source of mutual

dependence between capacity and demand is effective damping. As a building yield in

response to seismic demand it dissipates energy with hysteretic damping.

The capacity spectrum method initially characterizes seismic demand using an elastic

response spectrum. This spectrum is plotted in spectral ordinates (ADRS) format showing

the spectral acceleration as a function of spectral displacement. This format allows the

28

demand spectrum to be “overlaid” on the capacity spectrum for the building. The

intersection of the demand and capacity spectra, if located in the linear range of the

capacity, would define the actual displacement for the structure; however this is not

normally the case as most analyses include some inelastic nonlinear behavior. To find the

point where demand and capacity are equal, a point on the capacity spectrum need to be

selected as an initial estimate. Using the spectral acceleration and displacement defined by

this point, reduction factors may be calculated to apply to the 5% elastic spectrum to

account for the hysteretic energy dissipation, or effective damping, associated with the

specific point. If the reduced demand spectrum intersects the capacity spectrum at or near

the initial assumed point, then it is the solution for the unique point where capacity equals

demand. If the intersection is not reasonably close to the initial point, then a new point

somewhere between may be assumed and repeat the process until a solution is reached.

This is the performance point where the capacity of the structure matches the demand or

the specific earthquake.

Fig. 2.10 Determination of performance point (Adopted from ATC 40)

29

Once the performance point has been determined, the acceptability of a rehabilitation

design to meet the project performance objectives can be judged by evaluating where the

performance points falls on the capacity curve. For the structure and earthquake

represented by the overlay indicated in Fig. 2.10, the performance point occurs within the

central portion of the damage control performance range as shown in Fig. 2.9, indicating

that for this earthquake this structure would have less damage than permitted for the Life

Safety level and more than would be permitted for the Immediate Occupancy level. With is

information, the performance objective and/or the effectiveness of the particular

rehabilitation strategy to achieve the project performance objectives can be judged.

There are three procedures (A, B, C) to find performance point. Which procedure has to be

selected for analysis can be judged from the following comparison of three procedures.

Procedure A:

• Clearest, most transparent and direct application of the methodology

• Analytical method

• Conventional for spreadsheet programming

• May be the best method for beginners because it is most direct and easiest

to understand.

Procedure B:

• Analytical method

• Simpler than procedure A because of simplifying assumptions (that may

not always be valid)

• Most conventional for spreadsheet programming

• Reasonably transparent application of methodology

• Users of this method should fully understand the inherent assumptions

Procedure C:

• Graphical method

• Most convenient method for hand analysis

• Not as convenient for spreadsheet programming

• Least transparent application of methodology

30

2.4.14 Displacement coefficient method

Another procedure for calculating demand displacement is ‘Displacement Coefficient

Method’ which provides a direct numerical process for calculating the displacement

demand. Displacement Coefficient Method has not been explored. Performance analysis of

the structures under this thesis was made using Capacity Spectrum Method.

2.4.15 Seismic performance evaluation

The essence of virtually all seismic evaluation procedures is a comparison between some

measures of the “demand” that earthquake place on structure to a measure of the “capacity”

of the building to resist the induced effects. Traditional design procedures characterize

demand and capacity as forces. Base shear (total horizontal force at the lowest level of the

building) is the normal parameter that is used for this purpose. The base shear demand that

would be generated by a given earthquake, or intensity of ground motion is calculated, and

compares this to the base shear capacity of the building. If the building were subjected to a

force equal to its base shear capacity some deformation and yielding might occur in some

structural elements, but the building would not collapse or reach an otherwise undesirable

overall level of damage. If the demand generated by the earthquake is less than the capacity

then the design is deemed acceptable.

The first formal seismic design procedures recognized that the earthquake accelerations

would generate forces proportional to the weight of the building. Over the years, empirical

knowledge about the actual behavior of real structures in earthquakes and theoretical

understanding of structural dynamics advanced. The basic procedure was modified to

reflect the fact that the demand generated by the earthquake accelerations was also a

function of the stiffness of the structure.

The inherently better behavior of some buildings over others are also begun to recognize.

Consequently, that reduced seismic demand has been assumed for some structure based on

the characteristics of the basic structural material and system. The motivation to reduce

seismic demand for design came because it could not be rationalized theoretically how

structures resisted the forces generated by earthquakes. This was partially the result of the

31

fundamental assumption that structures resisted loads linearly without yielding or

permanent structural deformation.

2.4.16 Nonlinear static procedure for capacity evaluation of structures

Instead of comparing forces, nonlinear static procedures use displacements to compare

seismic demand to the capacity of a structure. This approach included consideration of the

ductility of the structure on an element by element basis. The inelastic capacity of a

building is then a measure of its ability to dissipate earthquake energy. The current trend in

seismic analysis is toward these simplified inelastic procedures.

The recommended central methodology is on the formulation of inelastic capacity curve for

the structure. This curve is a plot of the horizontal movement of a structure as it is pushed

to one side. Initially the plot is a straight line as the structure moves linearly. As the parts of

the structure yield the plot begins to curve as the structure softens. This curve is generated

by building a model of the entire structure from nonlinear representations of all of its

elements and components. Most often this is accomplished with a computer and structural

analysis software. The forces and displacement characteristics are specified for each piece

of the structure resisting the earthquake demand. These pieces are assembled geometrically

to represent the complete lateral load resisting system. The resulting model is subjected to

increasing increment of load in a pattern determined by its dynamic properties. The

corresponding displacements define the inelastic capacity curve of the building. The

generation of the capacity curve defines the capacity of the building uniquely and

independently of any specific seismic demand. In this sense it replaces the base shear

capacity of conventional procedures. When an earthquake displaces the building laterally,

its response is represented by a point on this curve. A point on the curve defines a specific

damage state of the building, since the deformation of its entire components can related to

the global displacement of the structure.



from the capacity curve itself. As the demand increases the structure eventually yields and,

32

as its stiffness decreases, its period lengthens. Since the seismic accelerations depend on

period, demand also changes as the structure yields. Another source of mutual dependence

between capacity and demand is effective damping. As building yields in response to

seismic demand, it dissipates energy with hysteretic damping. Building that have large,

stable hysteretic loops during cyclic yielding dissipate more than those with pinched loops

cause by degradation of strength and stiffness. Since the energy that is dissipated need not

be stored in the structure, the damping has the effect of diminishing displacement demand.

2.4.17 Structural performance levels and ranges

The Performance of a building under any particular event is dependent on a wide range of

parameters. These parameters are defined (ATC-40, 1996; FEMA 356, 2000) qualitatively

in terms of the safety afforded by the building to the occupants during and after the event;

the cost and feasibility of restoring the building to pre-earthquake condition; the length of

time the building is removed from service to effect repairs; and economic, architectural, or

historic impacts on the larger community. These performance characteristics are directly

related to the extent of damage that would be sustained by the building.

The federal emergency management agency in its report ‘prestandard and commentary for

the seismic rehabilitation of buildings (FEMA-356, 2000) defines the structural

performance level of a building to be selected from four discrete structural performance

levels and two intermediate structural performance ranges. The discrete Structural

Performance Levels are

Immediate Occupancy (S-1), Life Safety (S-3), Collapse Prevention (S-5), and

Not Considered (S-6). The intermediate Structural Performance Ranges are the

Damage Control Range (S-2) and the Limited Safety Range (S-4)

The definition of these performance ranges are given by FEMA [4]. Acceptance criteria for

performance within the damage control structural performance range may be obtained by

interpolating the acceptance criteria provided for the Immediate Occupancy (IO) and Life

Safety (LS) structural performance Levels. Acceptance criteria for performance within the

Limited Safety Structural Performance Range may be obtained by interpolating the

acceptance criteria provided for the life safety and collapse prevention structural

33

performance levels. The performance levels and ranges, as per FEMA are described is the

sections that follow.

2.4.17.1 Immediate occupancy structural performance level (S-1)

Structural performance level S-1, immediate occupancy, may be defined as the post-

earthquake damage state of a structure that remains safe to occupy, essentially retains the

pre-earthquake design strength and stiffness of the structure, and is in compliance with the

acceptance criteria specified in this standard for this structural performance levels defined

in Table 2-B1 to 2-B3 in the appendix.

Structural performance level S-1, immediate occupancy, means the post-earthquake

damage state in which only very limited structural damage has occurred. The basic vertical

and lateral-force-resisting systems of the building retain nearly all of their pre-earthquake

strength and stiffness. The risk of life-threatening injury as a result of structural damage is

very low, and although some minor structural repairs may be appropriate, these would

generally not be required prior to re-occupancy.

2.4.17.2 Damage control structural performance range (S-2)

Structural performance range S-2, damage control, may be defined as the continuous range

of damage states between the life safety structural performance level (S-3) and the

immediate occupancy structural performance level (S-1) defined in Table 2-B1 to 2-B3 in

the appendix.

Design for the damage control structural performance range may be desirable to minimize

repair time and operation interruption, as a partial means of protecting valuable equipment

and contents, or to preserve important historic features when the cost of design for

immediate occupancy is excessive.

2.4.17.3 Life safety structural performance level (S-3)

Structural performance level S-3, life safety, may be defined as the post-earthquake

damage state that includes damage to structural components but retains a margin against

34

onset of partial or total collapse in compliance with the acceptance criteria specified in

FEMA (FEMA-356, 2000) for this structural performance level defined in Table 2-B1 to 2-

B3 in the appendix.

Structural performance level S-3, life safety, means the post-earthquake damage state in

which significant damage to the structure has occurred, but some margin against either

partial or total structural collapse remains. Some structural elements and components are

severely damaged, but this has not resulted in large falling debris hazards, either within or

outside the building. Injuries may occur during the earthquake; however, the overall risk of

life-threatening injury as a result of structural damage is expected to be low. It should be

possible to repair the structure; however, for economic reasons this may not be practical.

While the damaged structure is not an imminent collapse risk, it would be prudent to

implement structural repairs or install temporary bracing prior to re-occupancy.

2.4.17.4 Limited safety structural performance range (S-4)

Structural performance range S-4, limited safety, may be defined as the continuous range of

damage states between the life safety structural performance level (S-3) and the collapse

prevention structural performance level (S-5) defined in Table 2-B1 to 2-B3 in the

appendix.

2.4.17.5 Collapse prevention structural performance level (S-5)

Structural performance level S-5, collapse prevention, may be defined as the post-

earthquake damage state that includes damage to structural components such that the

structure continues to support gravity loads but retains no margin against collapse in

compliance with the acceptance criteria specified FEMA for this structural performance

level defined in Table 2-B1 to 2-B3 in the appendix.

Structural performance level S-5, collapse prevention, means the post-earthquake damage

state in which the building is on the verge of partial or total collapse. Substantial damage to

the structure has occurred, potentially including significant degradation in the stiffness and

strength of the lateral-force resisting system, large permanent lateral deformation of the

35

structure, and to more limited extent degradation in vertical load carrying capacity.

However, all significant components of the gravity load resisting system must continue to

carry their gravity load demands. Significant risk of injury due to falling hazards from

structural debris may exist. The structure may not be technically practical to repair and is

not safe for re-occupancy, as aftershock activity could induce collapse.

2.4.17.6 Target building performance levels

Building performance is a combination of the both structural and nonstructural

components. Table 2-B1, 2-B2 and 2-B3 (FEMA-356) describe the approximate limiting

levels of structural damage that may be expected of buildings evaluated to the levels

defined for a target seismic demand. These tables represent the physical states of

mathematical calculation of different performance levels.

2.4.18 Response limit

To determine whether a building meets a specified performance objective, response

quantities from a nonlinear analysis are compared with limits given for appropriate

performance levels (ATC-40 and FEMA-356). The response limits fall into two categories:

2.4.18.1 Global building acceptability limits

These response limits include requirements for the vertical load capacity, lateral load

resistance, and lateral drift. Table 2.10 gives the limiting values for different performance

level.

Gravity loads

The gravity load capacity of the building structure must remain intact for acceptable

performance at any level. Where an element or component loses capacity to support gravity

loads, the structure must be capable to redistributing its load to other elements or

components of the existing system.

Lateral loads

Some components types are subjected to degrading over multiple load cycles. If a

significant number of components degrade, the overall lateral force resistance of the

36

building may be affected. The lateral load resistance of the building system, including

resistance to the effects of gravity loads acting through lateral displacements, should not

degrade by more than 20 percent of the maximum resistance of the structure for the

extreme case.

Two effects can lead to loss of lateral load resistance with increasing displacement. The

first is gravity loads acting through lateral displacements, known as the P-∆ effect. The P-∆

effect is most prominent for flexible structures with little redundancy and low lateral load

strength relative to the structure weight. The second effect is degradation in resistance of

individual components of the structure under the action of reversed deformation cycles.

When lateral load resistance of the building degrades with increasing displacement, there is

a tendency for displacements to accumulate in one direction. This tendency is especially

important for long-duration events. The following table presents deformation limits of

various performance levels. Maximum total drift is defined as the inter-story drift at the

performance point displacement. Maximum inelastic drift is defined as the portion of the

maximum total drift beyond the effective yield point. For Structural Stability, the

maximum total drift in story i at the performance point should not exceed the quantity

0.33Vi/Pi, where Vi is the total calculated shear force in story i and Pi is the total gravity

load(i.e. dead plus likely live load) at story i(ATC-40).

Table 2.10 Deformation limits (ATC-40)

Performance Level

Inter story drift limit Immediate

occupancy

Damage

control

Life

safety

Structural

stability

Maximum total drift 0.01 0.01 ~0.02 0.02 i

i

PV

33.0

Maximum inelastic

drift

0.005 0.005~0.015 No limit No limit

37

2.4.18.2 Element and component acceptability limit

Deformation and force controlled actions

Fig. 2.11 Component force versus deformation curves (FEMA-356)

All structural actions may be classified as either deformation controlled or force-controlled

using the component force versus deformation curves shown in Fig. 2.11 The Type 1 curve

depicted in Fig. 2.11 is representative of ductile behavior where there is an elastic range

(point 0 to point 1 on the curve) followed by a plastic range (points 1 to 3) with non-

negligible residual strength and ability to support gravity loads at point 3. The plastic range

includes a strain hardening or softening range (points 1 to 2) and a strength-degraded range

(points 2 to 3). Primary component actions exhibiting this behavior shall be classified as

deformation-controlled if the strain-hardening or strain-softening range is such that e > 2g;

otherwise, they shall be classified as force controlled. Secondary component actions

exhibiting Type 1 behavior shall be classified as deformation-controlled for any e/g ratio.

The Type 2 curve depicted in Fig. 2.11 is representative of ductile behavior where there is

an elastic range (point 0 to point 1 on the curve) and a plastic range (points 1 to 2) followed

by loss of strength and loss of ability to support gravity loads beyond point 2. Primary and

secondary component actions exhibiting this type of behavior shall be classified as

deformation-controlled if the plastic range is such that e >2g; otherwise, they shall be

classified as force controlled. The Type 3 curve depicted in Fig. 2.11 is representative of a

brittle or non-ductile behavior where there is an elastic range (point 0 to point 1 on the

curve) followed by loss of strength and loss of ability to support gravity loads beyond

38

point1. Primary and secondary component actions displaying Type 3 behavior shall be

classified as force-controlled (FEMA-356).

Deformation controlled and force controlled behavior

Acceptance criteria for primary components that exhibit Type 1 behavior are typically

within the elastic or plastic ranges between points 0 and 2, depending on the performance

level. Acceptance criteria for secondary elements that exhibit Type 1 behavior can be

within any of the performance ranges. Acceptance criteria for primary and secondary

components exhibiting Type 2 behavior will be within the elastic or plastic ranges,

depending on the performance level. Acceptance criteria for primary and secondary

components exhibiting Type 3 behavior will always be within the elastic range. Table 2.11

provides some examples of possible deformation- and force-controlled actions in common

framing systems.

Table: 2.11 Examples of possible deformation controlled and force controlled actions

(FEMA)

Component Deformation controlled

action

Force controlled action

Moment frames

Beam

Columns

Joints

Moment(M)

M

-

Shear (V)

Axial load (P), V

V1

Shear Walls M, V P

Braced Frames

Braces

Beams

Columns

Shear Link

P

-

-

V

-

P

P

P, M

Connections P, V, M3 P, V, M

Diaphragms M, V2 P, V, M

39

2.4.19 Acceptability limit

A given component may have a combination of both force and deformation controlled

actions. Each element must be checked to determine whether its individual components

satisfy acceptability requirements under performance point forces and deformations.

Together with the global requirements, acceptability limits for individual components are

the main criteria for assessing the calculated building response.

d0.75d

A

B

C

D E

∆

Late

ral l

oad

Lateral deformation∆ y

Q/Q

c

1.0

Life safety performance level

Structural stabilityperformance level

Fig. 2.12 Force deformation action and acceptance criteria (ATC-40)

The Fig. 2.12 shows a generalized load deformation relation appropriate for most concrete

components. The relation is described by linear response from A (unloaded component) to

an effective yield point B, linear response at reduced stiffness from B to C, sudden

reduction in lateral load resistance to D, response at reduced resistance to E, and final loss

of resistance thereafter. The following main points relate to the depicted load-deformation

relation:

a) Point A corresponds to the unloaded condition. The analysis must recognize

that gravity loads may induce initial forces and deformations that should be

accounted for in the model. Therefore, lateral loading may commence at a point

other than the origin of the load-deformation relation.

b) Point B has resistance equal to the nominal yield strength. The slope from B to

C, ignoring the effects of gravity loads acting through lateral displacements, is

usually taken as between 5% and 10% of the initial slope. This strain hardening,

40

which is observed for most reinforced concrete component, may have an

important effect on the redistribution of internal forces among adjacent

components.

c) The abscissa at C corresponding to the deformation at which significant

strength degradation begins.

d) The drop in resistance from C to D represents initial failure of the component.

e) The residual resistance from D to E may be non-zero in some cases and may be

effectively zero in others. Where specific information is not available, the

residual resistance usually may be assumed to be equal to 20% of the nominal

strength.

f) Point E is a point defining the maximum deformation capacity. Deformation

beyond that limit is not permitted because gravity load can no longer be

sustained.

Table 2-C1 to 2-C7 (Appendix) give the acceptance criteria (ATC-40) to be used with

Nonlinear Procedures for the acceptance model of individual structural elements of a

structure that has been used to evaluated for finding seismic performance of the selected

buildings under this thesis.

2.5 Nonlinear time history analysis

Earthquake excitation is time dependent, highly irregular and arbitrary in nature. Usually

earthquake excitation in the form of acceleration or displacement or velocity is recorded for

a time interval of 0.02 to 0.005 seconds. In this dynamic analysis procedure the response of

a structure at every time interval is recorded for the whole earthquake period and the

statistical average is represented. Because of its inherent complexities of the procedure and

nondeterministic nature of the input ground motion, the analysis procedure has not become

popular in the design houses for designing of the structures.

Ground motion time history developed for the specific site shall be representative of the

actual earthquake motion for an earthquake. When this procedure is followed an elastic or

inelastic dynamic analysis of a structure shall be made using a mathematical model of a

structure and applying at its base or any other appropriate level, a ground motion time

41

history. Time dependent dynamic response of the structure shall be obtained through

numerical integration of its equation of motion.

2.5.1 Nonlinear dynamic analysis for earthquake ground motions:

Earthquake causes vibration of the ground which is primarily a horizontal movement,

although some vertical movement is also present. The vibrations are time-dependent

typically for durations of 10-40 seconds, which increases gradually to the peak amplitude

and then decays. The ground vibration is expressed through the temporal variation of

ground accelerations which is used in structural analyses. Figs. 4.1 and 4.2 shows the

ground accelerations recorded during some of the best known and widely studied

earthquakes of the 20th century which are also used in the present study.

The behavior of RC under dynamic loading is not linear when the deformation is large and

load is time dependent. The use of linearly elastic analysis procedure is not valid in such

cases. In fact there are some situations where the use of such simplified analyses can be

misleading and missing in important details. The material and geometric properties which

are considered constant in linear analysis do not remain constant in many practical

situations. For example severe earthquake vibrations may cause quite large structural

deformations and as a result alter the stiffness properties significantly. Moreover, member

properties like mass or damping may undergo changes during the dynamic response, while

stiffness properties may vary significantly due to the material and geometric nonlinearities

caused by significant axial forces. Modal analysis is a commonly used method of linear

dynamic analysis, but is not valid for nonlinear systems. The incremental numerical

scheme needs to be applied for the dynamic analysis of nonlinear systems like Reinforced

Concrete structures.

2.5.2 Linearly elastic and inelastic systems:

For a linearly elastic system, the relationship between the applied force fs and the resulting

deformation u is linear; i.e.,

fs = k u ……………………………… (c)

42

where k is the linear stiffness of the system; its units are force/length. Here the resisting

force is directly proportional to the displacement and is a single valued function of u.

This is however not valid when the load-deformation relationship is nonlinear, i.e., when

the stiffness itself is not constant but is a function of u. Moreover, structural components

undergo cyclic deformation for dynamic problems. The initial loading curve is nonlinear at

the amplitudes of deformation; the unloading and reloading curves differ from the initial

loading path. This implies that the force fs corresponding to deformation u is not single-

valued and depends on the history of the deformations. It further depends on the rate of

change of deformation (i.e., the velocity, particularly on whether it is positive or negative).

Thus the resisting force for nonlinear dynamic problems can be expressed as

fs = fs(u,v) ………………………….(d)

and the system is called inelastic dynamic system.

For RC structures undergoing large deformations due to strong earthquake vibrations, Eq.

(d) is more appropriate than Eq. (c). The use of linear elastic dynamic analysis for strong

seismic problems can lead to the omission of such important concepts as plasticity,

yielding, shifting of equilibrium position, permanent deformation and residual stresses.

2.5.3 Equation of motion for seismic vibration

The governing equation of motion for an inelastic SDOF system subjected to ground

motion ug(t) is given by

m d2u/dt2 + c du/dt + k u = c dug/dt + k ug ……...…..…………...(e)

⇒ m d2ur/dt2 + c dur/dt + k ur = −m d2ug/dt2 ……..…..…..…………..(f)

where ur = u− ug is the relative displacement of the SDOF system with respect to the

ground displacement. Eq. (e) shows that the ground motion appears on the right side of the

equation of motion just like a time-dependent load. Therefore, although there is no body-

force on the system, it is still subjected to dynamic excitation by the ground displacement.

43

2.5.4 Solution by incremental time step integration:

For an inelastic SDOF system the equation of motion to be solved numerically is

m d2u/dt2 + c du/dt + fs(u,v) = f(t) …………………… (g)

subject to specified initial conditions. In Eq. (g) the system is assumed to have linear

viscous damping, but other forms of damping, including nonlinear damping could be

considered.

The applied force f(t) is given by a set of discrete values, fi = f(t), i = 0~ N. The time

interval, ∆ti = ti+1− ti is usually taken to be a constant. The response is determined at the

discrete time instants ti. The displacement, velocity and acceleration of the SDOF system

at time ti are ui, vi and ai and at time ti+1 are ui+1, vi+1 and ai+1 respectively. These values,

satisfy Eq. (g) at time ti and ti+1

m ai + c vi + (fs)i = fi ……………………………………. (h)

m ai+1 + c vi+1 + (fs)i+1 = fi+1 ………………………………(i)

When applied successively with i = 0, 1, 2,…..... the time stepping procedure gives the

desired response at all-time instances i = 1, 2, 3,...... The known initial conditions provide

the information necessary to start the procedure. The difference between Eq. (h) and (i)

gives an incremental equilibrium equation

m ∆ai + c ∆vi + (∆fs)i = ∆fi ………….……….………….. (j)

The incremental resisting force, (∆fs)i = (ki)sec ∆ui ….……….………….(k)

where the secant stiffness (ki)sec, as shown in Fig. (d), cannot be determined because ui+1

is unknown. On the assumption that the secant stiffness (ki)sec could be replaced by the

tangent stiffness (ki)T [as shown in Fig.(d)] over a small time step ∆t, Eq. (j) is

approximated by

(∆fs)i = (ki)T ∆ui ……………………….……….………...(l)

Dropping the subscript T from (ki)T in Eq. (l) and substituting it in Eq. (j) gives

m ∆ai + c ∆vi + ki ∆ui = ∆fi …………………...……..….(m)

44

fs

(fs)i+1 (ki)T

(∆fs)i (ki)sec

(fs)i

∆ui

ui ui+1 u

Fig. 2.13 Nonlinear force-displacement relationship

The similarity between this equation and the corresponding equation for linear systems

suggests that the non-iterative formulation used for linear systems may also be used in the

analysis of nonlinear response. It is necessary only to replace k by the tangent stiffness ki

to be evaluated at the beginning of each time step. In this work, the other properties (like

m and c) are assumed constant for each step.

2.5.5 The average acceleration method

Many approximate methods are possible to implement the numerical scheme (described

above) efficiently. Convergence, stability and accuracy are the three important

requirements for a numerical procedure. The Average Acceleration Method (special case of

the Newmark-β method), known for its simplicity and unconditional stability for linear

systems, can be used here replacing k by the tangent stiffness ki to be evaluated at the

beginning of each time step.

In the incremental formulation, some adjustments are made to the original formulations for

linear dynamic analysis. Using incremental displacement ∆ui (= ui+1−ui) and incremental

force ∆fi (= fi+1−fi), the incremental time-step integration shown below can be followed.

45

The simulation should start with two initial conditions, like the displacement u0 and velocity v0 at time t0 = 0. The initial acceleration can be obtained from the equation of motion at time t0 = 0 as

a0 = (f0 − cv0 − ku0)/m

The incremental equations used in this formulation are

(ki + 2c/∆t + 4m/∆t2) ∆ui = ∆fi + (2c + 4m/∆t) vi + (2m) ai

ui+1 = ui +∆ui

ai+1 = 4 ∆ui/∆t2 − 4vi/∆t − ai

vi+1 = vi + (ai + ai+1)∆t/2

Once the incremental displacement ∆ui is obtained from above equation, it can be used to

calculate the total displacement (ui+1), acceleration (ai+1) and velocity (vi+1) from above

equation.

2.9 Conclusion

In this chapter pushover analysis and earthquake design philosophy have been explained.

There are different earthquake analysis methods which have been discussed here. In

between those the procedure of finding earthquake force in a structure from BNBC [1] is

illustrated. The pushover analysis to find the actual performance of the structure is

discussed here in detail. Also dynamic analysis and time history analysis is discussed in

this chapter.

46

Chapter 3

Pushover Analysis

3.1 Introduction

An elastic analysis gives a good indication of the elastic response of structures, but it cannot predict

failure mechanisms and account for redistribution of forces during progressive yielding for an

earthquake excitation. Inelastic analyses procedures help demonstrate how buildings really work by

identifying modes of failure and potential for progressive collapse. The use of inelastic procedures for

design and evaluation is an approach to help engineers better understands how structures will behave

when subjected to major earthquakes, where it is assumed that the elastic capacity of the structure

will be exceeded. Application of this resolves some of uncertainties associated with code and elastic

procedures.

Various analysis methods are available, both linear and nonlinear for evaluation of concrete building.

The basic inelastic method is nonlinear time history analysis method. This method is too complicated

and considered impractical for general use. The central focus of this thesis is to study the simplified

nonlinear procedure for the generation of the “pushover” or capacity curve of a structure. This

represents the plot of progressive lateral displacement as a function of the increasing level of force

applied to the structure. Pushover analysis is a simplified static nonlinear analysis method which use

capacity curve and reduced response spectrum to estimate maximum displacement of a building

under a given level of earthquake.

In this chapter three reinforced concrete frames 2, 5, and 12 storied 2D frames are modeled. Two

dimensional models of case study frames are prepared using SAP2000 [21], ETABS [20] and

SeismoStruct [50] by considering the necessary geometric and strength characteristics of all members

that affect the nonlinear seismic response. The structural models are based on centerline dimensions

that beams and columns span between the nodes at the intersections of beam and column centerlines.

Rigid floor diaphragms are assigned at each story level and the seismic mass of the frames are

lumped at the mass center of each story. Gravity loads consisting of dead loads and 25% of live loads

are considered in pushover. The dynamic properties of the case study frames are summarized in Table

3.1-3.3. The configuration, member details and dynamic properties of case study frames are presented

in this chapter. Both pushover and nonlinear time history analyses are performed using gross section

properties and P-Delta effects are considered. Nonlinear member behavior of concrete sections is

modeled as discussed below for SAP2000 [21], ETABS [20] and SeismoStruct [50].

47

In this chapter the procedure for structural performance evaluation in the light of ATC-40[3] and

FEMA 356[4] has been described. For the performance evaluation purposes Dhaka is selected as the

site and seismic demand for Dhaka has been estimated as per guideline of ATC-40 and the frames are

designed as per the provisions of BNBC [1]. Structural performances of three 2D frames with

different configuration have been investigated. All structures have regular geometry and stiffness.

The performances of the structures as evaluated through pushover analysis have been presented

through capacity curves and capacity spectrums described in this chapter.

3.2 Details of pushover analysis in SAP2000 and ETABS

Different software is available for performing nonlinear analysis of concrete structure. In this study

ETABS [20] and SAP2000 [21] have been used. In this section, different features and options of

ETABS and SAP2000 related to perform pushover analysis is discussed. The structure should be

modeled develop according to the architectural design. The concrete and the steel property should be

provided according to design criteria. In ETABS and SAP2000 performance based analysis is suitable

for frame structure. The two types of loads (gravity and lateral) should be provided according to code

provisions. In this regard the loads have to be defined first and then have to be assigned on the

structure. In ETABS and SAP2000 superimposed dead loads have to be assigned and it takes

buildings own weight from the structure and its property. Live loads should be assigned according its

intended use. For seismic load UBC 94 should be used which satisfy BNBC [1].

After running analysis the structure should be designed from concrete frame design options. From

this the beam and column sections which are assigned first can also be checked. But the nonlinear

analysis takes the column sections which are design first. In defining column sections the

reinforcement to be designed option should be provided so that column reinforcement would be

checked when structure is designed.

Frame nonlinear hinge properties are used to define nonlinear force displacement and/or moment

rotation behavior that can be assigned to discrete locations along the length of frame (line) elements.

These nonlinear hinges are only used during static nonlinear (pushover) analysis. The hinge

properties are in ETABS and SAP2000 for concrete members and are generally based on in ATC-

40[3]. The hinge properties cannot be modified. They also can not be viewed because the default

properties are section dependent. The default properties can not be fully defined by the program until

the section to which they apply is identified. Thus, to see the effect of the default properties, the

48

default property should be assigned to a frame element, and then the resulting generated hinge

property should be viewed.

Fig. 3.1: Concrete moment and P-M-M hinge property

Slope between points B and C is taken as 10% total strain hardening for steel. Points C, D and E of

M3 based on Table A4 in appendix. The four conforming transverse reinforcing rows of the table are

averaged. My based on reinforcement provided, otherwise based on minimum allowable

reinforcement. P-M-M curve is for M3 (major moment) and is taken to be the same as the moment

curve in conjunction with the definition of axial–moment interaction curves. Points C, D and E of P-

M-M curve based on Table A5 in appendix. The four conforming transverse reinforcing rows of the

table are averaged.

Fig. 3.2 Concrete Shear hinge property

B C

D

1

COMPRESSION

TENSION

E

d

A

B- C-

D- E-

1.1

-1 -1.1

B C

D

1

COMPRESSION

TENSION

E

a

A

B-C-

D- E-

1.1

-1 -1.1

b

c

e

49

Slope between points B and C is taken as 10% total strain hardening for steel. Points C, D and E

based on Table 2 A1-A4 (ATC-40) of Appendix, Item 2, average of the two rows labeled

“Conventional longitudinal reinforcement with conforming transverse reinforcement” in appendix.

The acceptance criteria values are deformations (displacements, strains, or rotations) that have been

normalized by the same deformation scale factors used to specify the load deformation curve, and are

typically located between points B and C and points B’ and C’ on the curve. They are used to indicate

the state of the hinge when viewing the results of the analysis, but they do not affect the behavior of

the structure. These acceptance criteria of M3, V2 and P-M-M hinge are taken from Table 2 A1, A2,

A3 and A4 of appendix.

Static nonlinear analysis can consist of any number of cases. ETABS and SAP 2000 have been used

for the analysis purpose. Each static nonlinear case can have a different distribution of load on the

structure. A static nonlinear case may start from zero initial conditions, or it may start from the results

at the end of a previous case. Each analysis case may consist of multiple construction stages.

For static nonlinear analysis, displacement controlled is used. The load combination specified in the

load pattern area of the form is applied, but its magnitude is increased or decreased as necessary to

keep the control displacement increasing in magnitude. This option is useful for applying lateral load

to the structure, or for any case where the magnitude of the applied load is not known in advance, or

when the structure can be expected to lose strength or become unstable.

The conjugate displacement is a generalized displacement measure that is defined as the work

conjugate of the applied Load Pattern. It is a weighted sum of all displacement degrees of freedom in

the structure: each displacement component is multiplied by the load applied at that degree of

freedom, and the results are summed. The conjugate displacement is usually the most sensitive

measure of displacement in the structure under a given specified load. It is usually recommended that

to use the conjugate displacement unless one can identify a displacement in the structure that

monotonically increases during the analysis. The monitored displacement is a single displacement

component at a single point that is monitored during a static nonlinear analysis. When plotting the

pushover curve, the program always uses the monitored displacement for the horizontal axis. The

monitored displacement is also used to determine when to terminate a displacement controlled

analysis.

To start the current cases from the end condition of a previously specified static nonlinear case, select

the name of the previous case from the start from previous case drop down list. Typically this option

is used for a lateral static nonlinear case to specify that it should start from the end of a gravity static

50

nonlinear case. To get the positive displacement increments of the pushover curve to be saved, the

save positive increments only check box should be checked. In the following example, the solid line

represents the pushover curve if the save positive increments only check box is checked (the default)

and the dashed line represents the pushover curve if the save positive increments only check box is

not checked. Fig. 3.3 showed that positive increment and all increment are saved.

Fig. 3.3 An example to illustrate the option of Save Positive Increments Only

The minimum saved steps restricts the maximum step size used to apply the load in a static nonlinear

case. ETABS and SAP2000 automatically creates steps corresponding to events on the hinge stress-

strain curves or to significant nonlinear geometric effects. The results of these steps are saved only if

they correspond to a significant change in the slope of the pushover curve. The maximum null steps is

used, if necessary, to declare failure (i.e., non-convergence) in a run before it reaches the specified

force or displacement goal. The program may be unable to converge on a step when catastrophic

failure occurs in the structure, or when the load cannot be increased in a load-controlled analysis.

There may also be instances where it is unable to converge in a step because of numerical sensitivity

in the solution. The maximum total steps limits the total time the analysis will be allowed to run.

ETABS and SAP2000 attempts to apply as much of the specified load pattern as possible, but may be

restricted by the occurrence of an event, failure to converge within the maximum iterations/step, or a

limit on the maximum step size from the minimum number of saved steps. As a result, a typical static

nonlinear analysis may consist of a large number of steps. Additional steps may be required by some

member unloading methods to redistribute load.

0 10 20 30 40 50 60 70

0 2 4 6 8 10 12

positive increments are saved onlyall increments are saved

51

The large displacement option should be used for cable structures undergoing significant

deformation; and for buckling analysis, particularly for snap-through buckling and post-buckling

behavior. The frame elements and other elements that undergo significant relative rotations within the

element should be divided into smaller elements to satisfy the requirement that the strains and

rotations within an element are small. For most other structures, the P-delta option is adequate,

particularly when material nonlinearity dominates. If reasonable, it is recommended that the analysis

be performed first without P-delta (i.e., use none), adding geometric nonlinearity effects later

3.3 Description of case study frames for validation

The effects of lateral load patterns modes on global structural behavior and on the accuracy of

pushover predictions have been studied on reinforced concrete moment resisting frames. Three

reinforced concrete frames with 2, 5, and 12-stories are model in this chapter. Two dimensional

models of case study frames are prepared using SAP2000 [21] and ETABS [20] by considering the

necessary geometric and strength characteristics of all members that affect the nonlinear seismic

response. The structural modeled are based on centerline dimensions that beams and columns span

between the nodes at the intersections of beam and column centerlines and beam column joints are

not modeled. Rigid floor diaphragms are assigned at each story level and the seismic mass of the

frames are lumped at the mass center of each story. Gravity loads consisting of dead loads and 25%

of live loads are considered in pushover analyses. The free vibration analyses of the frames using

SAP2000 [21] and ETABS [20] yielded exactly same dynamic properties. The configuration,

member details and dynamic properties of case study frames are presented in Table 3.1 to 3.6. The

details of shear reinforcement are not considered since controlling behavior of frame members is

assumed to be flexure. The pushover analysis is performed using gross section properties and P-Delta

effects are neglected. Nonlinear member of concrete sections are modeled as discussed in this chapter

for SAP2000 and ETABS.

3.3.1 Two story RC frame

Two story frame is consist two bay moment resisting frame. It is fixed at its support. Typical floor

height is 3.962m. The length of each bay is 7.315m. Other structural dimensions and loading

properties are given in Table 3.1. Dynamic properties are given in Table 3.2.

52

Fig. 3.4 Two story RC Frame

Table 3.1 Cross section and loading properties of two story RC frame

Properties of beam

Story Beam Mass (t) DL (kN/m) LL (kN/m)

Dimension (mm) Reinforcement (sq mm)

Depth Width Top Bottom

1 558 304 1342 3148 178 25.71 1.05

2 508 304 1342 2503 98 19.23 0.98

Properties of column

Story Column Dimension (mm) Number of Bars Bar Area

(sq mm) X-dir Y-dir X-dir Y-dir

1 and 2 Exterior 609.6 609.6 5 5 645.16

Interior 609.6 609.6 5 5 645.16

Concrete and steel properties of 2-story RC frame

Property Concrete Steel

Ultimate strength (MPa) 26 494.4

Modulus of elasticity (MPa) 28730.5

53

Table 3.2 Dynamic properties of 2-story RC frame

Modal properties Mode

1 2

Period, T (sec) 0.488 0.148

Modal participation factor 1.336 0.336

Modal mass factor 0.834 0.166

3.3.2 Five story RC frame

Five story frame is consist two bay moment resisting frame. It is fixed at its support. Typical floor

height is 3.962m. The length of each bay is 7.315m. Other structural dimensions and loading


Fig. 3.5 Five story RC frame

54

Table 3.3 Cross section and loading properties of five story RC frame

Properties of beam

Story Beam Mass (t) DL (kN/m) LL (kN/m)

Dimension (mm) Reinforcement (sq mm )


1-4 660.4 406.4 5083.86 3148.38 104.025 20.49 1.31

5 508.0 304.8 3793.54 2503.22 77.056 15.64 0.53


Story Column Dimension (mm) Number of Bars Bar Area (sq mm)

X-dir Y-dir X-dir Y-dir

1-5 Exterior 711.2 711.2 6 6 885.8

Interior 711.2 711.2 6 6 885.8



Ultimate strength (MPa) 27.6 459.2


Table 3.4 Dynamic properties of 5-story RC frame


1 2 3

Period, T (sec) 0.857 0.272 0.141

Modal participation factor 1.348 0.528 0.258

Modal mass factor 0.794 0.116 0.054

3.3.3 Twelve story RC frame

Twelve story RC frame is consist four bay moment resisting frame. It is fixed at its support. Typical

floor height is 3.962m. The length of each bay is 7.315m. Other structural dimensions and loading


55

Fig. 3.6 Twelve story RC frame

Table 3.5 Cross Section and Loading properties of twelve story RC frame

Properties of beam

Story Beam Mass (t) DL (kN/m) LL

(kN/m) Dimension (mm) Reinforcement (sq mm )


1-3 1016 508.0 6625.79 6116.12 346.860 16.78 1.1

4-7 914.4 508.0 6625.79 6116.12 346.860 16.78 1.1

8-11 762.0 457.2 5096.76 4077.41 346.860 16.78 1.1

12 609.6 457.2 2038.71 1019.35 294.330 13.45 0.44

56


Story Column Dimension (mm) Number of Bars Bar Area

(sq mm) X-dir Y-dir X-dir Y-dir

1-3 Exterior 1219.2 1219.2 12 12 509.7

Interior 1524.0 1524.0 7 7 509.7

4-8 Exterior 1117.6 1117.6 8 8 509.7

Interior 1447.8 1447.8 6 6 509.7

9-12 Exterior 1016.0 1016.0 7 7 509.7

Interior 1270.0 609.6 5 5 509.7



Ultimate strength (MPa) 27.6 459.2


Table 3.6 Dynamic properties of twelve story RC frame


1 2 3

Period, T (sec) 1.610 0.574 0.310

Modal participation factor 1.398 0.615 0.372

Modal mass factor 0.730 0.130 0.052

3.4 Validation of pushover curve for 2D frame

Several types of output have been obtained from the static nonlinear analysis. Base reaction versus

monitored displacement has been plotted. Tabulated values of base reaction versus monitored

displacement at each point along the pushover curves, along with tabulations of the number of hinges

beyond certain control points on their hinge property force displacement curve has been plotted in

different figures. Base reaction versus monitored displacement has been plotted in figures. Pushover

curves are obtained by performing pushover analyses using SAP2000 [21] and ETABS [20]. The

base shear and story displacement data extracted from pushover analysis using SAP2000 [21],

ETABS [20] are plotted in Fig. 3.7, 3.8 and 3.9. The base shear and story displacement data extracted

from Oguz [13] which are superimposed in Figs. 3.7, 3.8 and 3.9. The superimposed pushover curves

are same in elastic part but inelastic part is minor differences. The minor differences between

57

SAP2000 [21] and ETABS [20] can be attributed to assumption and simplification of modeling. From

Fig.3.7, 3.8 and 3.9, it is seen that pushover curves are almost same. So it has been concluded that

nonlinear pushover analysis using SAP2000 [21] and ETABS [20] are close to same with results of

Oguz [13].

Fig. 3.7 Pushover curve for 2-story 2D frame

Fig. 3.8 Pushover curve of the 5-story 2D frame

Fig. 3.9 Pushover curve of the 12-story 2D frame

58

3.5 Comparison of pushover curve for SAP2000 and SeismoStruct

3.5.1 Description of SeismoStruct

SeismoStruct [50] is a finite element package capable of predicting the large displacement behavior

of space frames under static or dynamic loading, taking into account both geometric nonlinearities

and material inelasticity. Concrete, steel material models are available, together with a large library

of 3D elements that may be used with a wide variety of pre-defined steel, concrete and composite

section configurations. With the wizard facility the user can create regular/irregular 2D or 3D

models and run all types of analyses. The whole process takes no more than a few seconds. Seven

different types of analysis are as follows, eigenvalue analysis, static analysis (non-variable load),

static pushover analysis, static adaptive pushover analysis, static time history analysis, dynamic

time history analysis, incremental dynamic analysis (IDA). conventional (non-adaptive) pushover

analysis is frequently utilized to estimate the horizontal capacity of structures featuring a dynamic

response that is not significantly affected by the levels of deformation incurred (i.e. the horizontal

load pattern, which aims at simulating dynamic response, can be assumed as constant). In static

time history analysis, the applied loads (displacement, forces or a combination of both) can vary

independently in the pseudo time domain, according to a prescribed load pattern. Dynamic analysis

is commonly used to predict the nonlinear inelastic response of a structure subjected to earthquake

loading (evidently, linear elastic dynamic response can also be modeled for as long as elastic

elements and/or low levels of input excitation are considered). The direct integration of the

equations of motion is accomplished using the numerically dissipative integration algorithm, a

special case of the former, the well-known Newmark scheme [39], with automatic time step

adjustment for optimum accuracy and efficiency (see automatic adjustment of load increment or

time step).

The applied loading may consist of constant or variable forces, displacements and accelerations at

the nodes. The variable loads can vary proportionally or independently in the pseudo time or time

domain. The program accounts for both material inelasticity and geometric nonlinearity. A large

variety of reinforced concrete, steel and composite sections are available. In the pushover method

the lateral load distribution is not kept constant but is continuously updated, according to the modal

shapes and participation factors derived by eigenvalue analysis carried out at the current step. In

this way, the stiffness state and the period elongation of the structure at each step, as well as higher

mode effects, are accounted for. SeismoStruct possesses the ability to smartly subdivide the loading

increment, whenever convergence problems arise. The level of subdivision depends on the

59

convergence difficulties encountered. When convergence difficulties are overcome, the program

automatically increases the loading increment back to its original value. SeismoStruct's processor

features real time plotting of displacement curves and deformed shape of the structure, together

with the ability of pausing and restarting the analysis. Performance criteria can also be set, allowing

the user to identify the instants at which different performance limit states (e.g. non structural

damage, structural damage, collapse) are reached. The sequence of cracking, yielding, failure of

members throughout the structure can also be, in this manner readily obtained.

Large displacements/rotations and large independent deformations relative to the frame element's

chord (also known as P-Delta effects) are taken into account in SeismoStruct. Hence, in

SeismoStruct, all analyses (with the obvious exception of eigenvalue procedures) are treated as

potentially nonlinear, implying the use of an incremental iterative solution procedure whereby loads

are applied in pre-defined increments, equilibrated through an iterative procedure.

Modeling of seismic action is achieved by introducing acceleration loading curves (accelerograms)

at the supports, noting that different curves can be introduced at each support, thus allowing for

representation of asynchronous ground excitation. In addition, dynamic analysis may also be

employed for modeling of pulse loading cases (e.g. blast, impact, etc.), in which case instead of

acceleration time histories at the supports, force pulse functions of any given shape (rectangular,

triangular, parabolic, and so on), can be employed to describe the transient loading applied to the

appropriate nodes. Currently, thirteen material types are available in SeismoStruct. By making use

of these material types, the user is able to create an unlimited number of different materials, used to

define the cross-sections of structural members. Materials that are to be available within a

SeismoStruct project come defined in the materials module, where the name (used to identify the

material within the project), type (listed above) and mechanical properties (i.e. strength, modulus of

elasticity, strain-hardening, etc.) of each particular material can be defined. Currently, twenty one

section types are available in SeismoStruct. These range from simple single material solid sections

to more complex reinforced concrete and composite sections. By making use of these section types,

the user is able to create up to 500 different cross sections, used to define the different element

classes of a structural model. rectangular solid section, rectangular hollow section, circular solid

section, circular hollow section, reinforced concrete rectangular section, reinforced concrete

circular section etc.

The different elements of the structure are defined in the element connectivity module, where their

name, element class and corresponding nodes are identified.

60

Constrain certain degrees of freedom of slave nodes to a master node, by means of a rigid link. In

other words, the rotations of the slave node are equal to the rotations of the master node, whilst the

translations of the former are computed assuming a rigid lever arm connection with the latter. Both

master and slave nodes need to be defined for this constraint type, and the degrees of freedom to be

slaved to the master node (restraining conditions) have to be assigned. The boundary conditions of

a model are defined in the restraints module, where all structural nodes are listed and available for

selection and restraining against deformation in any of the six degrees of freedom.

There are four load categories in SeismoStruct [50]. These can be applied to any structural model.

These comprise all static loads that are permanently applied to the structure. They can be forces

(e.g. self-weight) or prescribed displacements (e.g. foundation settlement) applied at nodes. When

running an analysis, permanent loads are considered prior to any other type of load, and can be used

on all analysis types, with the exception of Eigenvalue analysis, where no loading is present. Note

that gravity loads should be applied downwards, for which reason they always feature a negative

value.

3.5.2 Description of case study frames

The effects of lateral load patterns on global structural behavior and on the accuracy of pushover

predictions are studied on 2-Storied and 5-Storied 2D reinforced concrete moment resisting frames.

Two dimensional models of case study frames are prepared using SAP2000 [21] and SeismoStruct

[50] by considering the necessary geometric and strength characteristics of all members that affect the

nonlinear seismic response. The structural models are based on centre line dimensions that beams and

columns span between the nodes at the intersections of beam and column centerlines and beam-

column joints are not modeled. Rigid floor diaphragms are assigned at each story level and the

seismic mass of the frames are lumped at the mass center of each story. The dynamic properties of the

case study frame are summarized in Table 3.2. The configuration, member details and dynamic

properties of case study frame is presented in Table 3.1-3.6. The description of frame is shown in

Figs. 3.4, 3.5 and 3.6. Nonlinear member behavior of RC sections is modeled with SAP2000 [21] and

SeismoStruct [50].

3.5.3 Analysis and results

Several types of output have been obtained from the static nonlinear analysis. Base reaction versus

monitored displacement has been plotted. Tabulated values of base reaction versus monitored

61

displacement at each point along the pushover curve, force displacement curve have been plotted in

Fig.3.13 and 3.14.

3.5.3.1 Two storied 2D frame

Pushover curves are obtained by performing pushover analyses using SeismoStruct [50] and

SAP2000 [21] for 2-story 2D frame.. The base reaction is used for plotting the pushover curve. It is

the resultant force reaction caused by the load pattern applied in the given static nonlinear case. Story

displacements, story pushover curves for any lateral load pattern are extracted from the pushover

analysis. The base shear and story displacement data extracted from pushover analysis using

SeismoStruct [50] and SAP200 [21] are plotted in Fig. 3.10. The superimposed pushover curves are

close to same. The superimposed pushover curves are same in elastic part but inelastic part is quite

differences. The differences between SAP2000 [21] and SeismoStruct [50] can be attributed to

different approach, assumption and simplification of modeling.

The pushover analyses using uniform lateral load pattern yielded capacity curves are higher than

those of the triangular lateral load patterns, for 2-Storied 2D frames considered. This is expected

because capacity curve is a function of the point of application of the resultant of lateral load as

well as the nonlinear structural characteristics.

Fig. 3.10 Pushover curve of the 2-story 2D frame for different loading pattern

62

3.5.3.2 Five storied 2D frame

Pushover curves are obtained by performing pushover analyses using SeismoStruct [50] and

SAP2000 [21] for 5-story 2D frame.. The base reaction is used for plotting the pushover curve. It is

the resultant force reaction caused by the load pattern applied in the given static nonlinear case. Story

displacements, story pushover curves for any lateral load pattern are extracted from the pushover

analysis. The base shear and story displacement data extracted from pushover analysis using

SeismoStruct [50] and SAP200 [21] are plotted in fig. 3.11. The superimposed pushover curves are

almost same. The superimposed pushover curves are same in elastic part but inelastic part is minor

differences. The minor differences between SAP2000 [21] and SeismoStruct [50] can be attributed to

assumption and simplification of modeling.

The pushover analyses using 'uniform' lateral load pattern yielded capacity curves are higher than

those of the triangular lateral load patterns, for 5-Storied 2D frames considered.


The results of the analyses for the two programs are superimposed on one plot of the base shear

versus roof displacement shown in Fig 3.10 and 3.11. This figure demonstrates that the two

responses are almost identical for 2D frame. The differences displayed in the plot have been

attributed to errors caused by different approaches to modeling the beam and column elements.

Therefore it can be concluded that nonlinear pushover analysis using SAP2000 [21] and

SeismoStruct [50] are almost same and identical. The pushover analyses using uniform lateral load

63

pattern capacity curves are higher than those of the triangular lateral load patterns for 2D frames

with both software. The pushover analyses using 'Uniform' lateral load pattern capacity curves are

more accurate with both softwares. Therefore, finally it is concluded that nonlinear pushover

analysis run in SeismoStruct is valid and accurate.

3.6 Different loading pattern for pushover curve

The accuracy of invariant lateral load patterns utilized in pushover analysis to predict the behavior

imposed on the structure due to nonlinear response are evaluated in this study. For this purpose,

global structure behavior, story displacements, inter-story drift ratios, story shears and plastic hinge

locations are selected as response parameters.

Pushover curves are obtained by performing pushover analyses using SAP2000 [21] and ETABS

[20]. Story displacements, inter story drift ratios, story pushover curves and plastic hinge locations

for any lateral load pattern are extracted from the pushover database at the predetermined

maximum roof displacement consistent with the deformation level considered and are compared

with absolute maximum values of exact response parameters obtained from nonlinear analysis.

Capacity curves (base shear versus roof displacement) are the load-displacement envelopes of the

structures and represent the global response of the structures. Capacity curves for case study frames

are obtained from the pushover analyses using lateral load patterns and are shown in Fig. 3.12-3.14

3.6.1 Two storied 2D frame


those of the triangular lateral load patterns. This is expected because capacity curve is a function of

the point of application of the resultant of lateral load as well as the nonlinear structural

characteristics.

64


3.6.2 Five storied 2D frame


those of the triangular lateral load patterns, 'elastic first mode' for 5-storied 2D frames considered.

This is expected because capacity curve is a function of the point of application of the resultant of

lateral load as well as the nonlinear structural characteristics.


3.6.3 Twelve storied 2D frame


those of the triangular lateral load patterns, 'elastic first mode' for 12-storied 2D frames considered.

65

This is expected because capacity curve is a function of the point of application of the resultant of

lateral load as well as the nonlinear structural characteristics.


3.7 Performance evaluation of structure

In this section two reinforced concrete frames 2 and 5 storied 2D frames are modeled. These frames

are designed as per the provisions of BNBC [1]. Considering live load 29.19kN/m2 and dead load

58.38kN/m2 is used floor level. Selfweight of the concrete members are considering unit weight of

concrete as 23.56kN/m3. As per BNBC seismic modification factor R=8 (IMRF) has been considered.

Performance point of any structure demand curve is required and demand curve can be generated

with SAP2000 [21]. But several parameters are required to generate the curves. In this section, those

parameters are defined and demand curve is plotted by SAP2000 [21]. The absolute maximum values

of roof displacements and base shear are determined for each deformation level to approximate a

dynamic capacity curve for the frames. The performance point is determined for serviceability

earthquake (SE), design earthquake (DE) and maximum earthquake (ME).

Both the capacity curve and the demand curve are plotted here in the same plotted area in ADRS

format. The performance point in spectral acceleration versus spectral displacement coordinates. The

effective period and effective damping at the performance point. The shape of the demand spectrum

with 5% damping is controlled by the values input in the seismic coefficient CA and CV. The

parameters are determined as follows.

Establishment demand spectra:

Location of the site: Dhaka city

66

Soil profile at the site: Soil type SC as per Table 2.5 when the soil properties are not known in

sufficient detail.

Earthquake source type: C, Near Source factor:>15 Km

Table 3.7 Calculation of CA Seismic Zone Factor,Z 0.15 As per BNBC/93 0.15 As per BNBC/93 0.15 As per BNBC/93 Earthquake Hazard Level, E 0.5 Design Earthquake 1 Max Earthquake 0.35 Serviceability Earthquake Factored E (E'X1.4) 0.7 1.4 0.49

Near-Source Factor 1 >15km, table 2.4 1 >15km, table 2.4 1 >15km, table 2.4 Shaking Intensity, ZEN 0.105 0.21 0.0735 For Soil Type SC, CA 0.126 From Table 2.5 0.249 From Table 2.5 0.088 From Table 2.5

Table 3.8 Calculation of CV Seismic Zone Factor,Z 0.15 As per BNBC 0.15 As per BNBC 0.15 As per BNBC Earthquake Hazard Level, E 0.5 Design Earthquake 1 Max Earthquake 0.35 Serviceability Earthquake Factored E (E'X1.4) 0.7 1.4 0.49 Near-Source Factor 1 >15km, table 2.4 1 >15km, table 2.4 1 >15km, table 2.4 Shaking Intensity, ZEN 0.105 0.21 0.0735 For Soil Type SC, CV 0.178 From Table 2.6 0.333 From Table 2.6 0.128 From Table 2.6

3.7.1 Two story 2D frame

From Fig 3.15 it is seen that at the performance point for Procedure A, spectral displacement is

9.40mm and spectral acceleration is 0.22g for SE, spectral displacement is 13.47mm and spectral

acceleration is 0.317g for DE and spectral displacement is 24.0mm and spectral acceleration is 0.37g

for ME. From Fig 3.16 it is seen that at the performance point for Procedure B, spectral displacement

is 9.34mm and spectral acceleration is 0.22g for SE. From Fig 3.17 it is seen that at the performance

point spectral displacement is 13.374mm and spectral acceleration is 0.315g for DE. From Fig 3.18 it

is seen that at the performance point spectral displacement is 23.84mm and spectral acceleration is

0.377g for ME.

Fig. 3.15 Capacity spectrum of the 2-story 2D frame (Procedure A)

67

Fig. 3.16 Capacity spectrum of the 2-story frame for SE (Procedure B)

Fig. 3.17 Capacity spectrum of the 2-story frame for DE (Procedure B)

68

Fig. 3.18 Capacity spectrum of the 2-story frame for ME (Procedure B)

From Procedure A (Fig.3.15) and Procedure B (Figs.3.16 to 3.18), it has seen that performance

point spectral displacement and spectral acceleration are close to same. So it has been concluded

that capacity spectrum of 2-story frame is close to same for Procedure A and B.

3.7.1.1 Local level performance

The observations have been made from the comparison of plastic hinge locations determined by

pushover analyses. Plastic hinges obtained from pushover analyses are generally different for each

frame. Hinge curve for 2-Story frame is shown in Figs.3.19 to 3.21. From these Figures at

performance point, hinges are in the range of B-IO. For three level of earthquake, no hinge is found

to cross the Immediate Occupancy (IO) limits. According ATC-40[3], 2-story frame satisfies local

criteria. Therefore it has been said that 2-story frame structure fulfills the performances at local

level for serviceability earthquake, design and maximum earthquake.

Fig. 3.19 Deformation of the 2-story frame at SE level

69

Fig. 3.20 Deformation of the 2-story frame at DE level

Fig. 3.21 Deformation of the 2-story frame at ME level

3.7.1.2 Global level performance

From the Fig. 3.22, it is seen that the performance point of structure has maximum story drift of

0.0046 at story level 2 for 2-story frame. This is less than allowable IO level 0.01 described in ATC

40[3]. This structure satisfies the requirement of ATC 40[3]. Therefore it has been said that the

structure fulfills the performance objective at gloval level for serviceability earthquake (SE), design

earthquake (DE) and maximum earth quake (ME).

70

.

Fig. 3.22 Story drift ratio at performance point of 2-Story 2D frame for different earthquake level

Note: IO=Immediate Occupancy, LS= Life Safety, CP= Collapse Prevention

3.7.2 Five story 2D frame

From Fig. 3.23 it is seen that at the performance point for Procedure A, spectral displacement is

24.0mm and spectral acceleration is 0.14g for SE, spectral displacement is 34.0mm and spectral

acceleration is 0.173g for DE and spectral displacement is 61.0mm and spectral acceleration is 0.24g

for ME, . From Fig 3.24 it is seen that at the performance point for Procedure B, spectral

displacement is 24.73mm and spectral acceleration is 0.141g for SE. From Fig 3.25 it is seen that at

the performance point spectral displacement is 33.95mm and spectral acceleration is 0.174g for DE.

From Fig 3.26 it is seen that at the performance point spectral displacement is 61.22mm and spectral

acceleration is 0.243g for ME.

Fig. 3.23Capacity spectrum of the 5-story 2D frame (Procedure A)

71

Fig. 3.24 Capacity spectrum of the 5-story frame for SE (Procedure B)

Fig. 3.25 Capacity spectrum of the 5-story frame for DE (Procedure B)

72

Fig. 3.26 Capacity spectrum of the 5-story frame for ME (Procedure B)

From Procedure A (Fig.3.23) and Procedure B (Figs.3.24to 3.26), it has seen that performance

point spectral displacement and spectral acceleration are close to same. So it has been concluded

that capacity spectrum of 5-story frame is close to same for Procedure A and B.


The observations have been made from the comparison of plastic hinge locations determined by

pushover analyses. Plastic hinges obtained from pushover analyses are generally different for each

frame. Hinge curve for 5-Story frame is shown in Figs.3.27 to 3.29. From these Figures at

performance point, hinges are in the range of B-IO. For three level of earthquake, no hinge is found

to cross the Immediate Occupancy (IO) limits. According ATC-40[3], 5-story frame satisfies local

criteria. Therefore it has been said that 5-story frame structure fulfills the performance at local level

for serviceability, design and maximum earthquake.

Fig. 3.27 Deformation of the 5-story frame at SE level

73

Fig. 3.28 Deformation of the 5-story frame at DE level

Fig. 3.29 Deformation of the 5-story frame at ME level


From the Fig. 3.30, it is seen that the performance point of structure has maximum story drift ratio

is 0.006219 for ME at story level 2 for 5-story level. This is less than allowable IO level 0.01

described in ATC 40[3]. So this structure satisfies the requirement of ATC 40[3]. Therefore it has

been said that the structure fulfills the performance objective at gloval level for serviceability

earthquake (SE), design earthquake (DE) and maximum earth quake (ME).

74

Fig. 3.30 Story drift ratio at performance point of 5-Story 2D frame for different earthquake level


3.8 Conclusion:

This chapter presented the process of performance based analysis and the way by which the results

can be used to predict local and global performance of structure for serviceability, design and

maximum earthquakes. SAP2000, ETABS and SeismoStruct softwares have been used in tandem

to observe the effect of load pattern on capacity curves. Validation of the simplified nonlinear static

analysis i.e. pushover analysis has been confirmed by comparison the capacity curves of 2D frames

with published numerical results of Oguz [13]. Building frames design as per the provisions of

BNBC has been modeled in SAP2000 to investigate their performances and they have been found

to satisfy the local and global performance criteria as per ATC-40 quite easily for three levels of

earthquakes. ATC-40 contains two methods of demand evaluation namely Procedure A and

Procedure B. ETABS and SAP2000 software include Procedure B in determining the demand. An

Excel has been developed incorporating the ATC-40 Procedure A method. The results show that

both Procedures A and B yield comparable demand in terms of spectral acceleration and spectral

displacement for the 2D frames analyzed.

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Chapter 4

Time history analysis

4.1 Introduction

In chapter 3, pushover analysis has been used to check the performance of different RC

frames. Nonlinear time history analysis is a more rigorous method of modeling response of a

structure and can also be used to determine the performance of structure due to seismic force.

The pushover analysis may be more convenient than the nonlinear time history analysis

because of computational time. The results shown previous chapter analysis took minutes

with the pushover analysis and several hours with the nonlinear time history analysis. For this

reason pushover analysis is more practical for use in the design office. But nonlinear time

history analysis is more accurate in predicting the building performance than the pushover

analysis.

The nonlinear response of structures is very sensitive to the structural modeling and ground

motion characteristics. Therefore, two ground motion records that accounts for uncertainties

and differences in severity, frequency and duration characteristics has been used to predict

the possible deformation modes of the structures for seismic performance evaluation

purposes. However, for simplicity, seismic demand prediction is generally performed by

pushover analysis which mostly utilizes smoothened response spectra. In this chapter, the

accuracy of capacity prediction using pushover analyses for various invariant lateral load

patterns is evaluated in comparison with time history capacity obtained using selected ground

motion excitations.

In this chapter, the response of case study frames are studied in the elastic and inelastic

deformation levels that are represented by peak roof displacements on the capacity

(pushover) curve of the frames. For each frame, the ground motion record is gradually scaled

to obtain the peak roof displacement versus base shear curve which is comparable to the

capacity curve of pushover analysis as shown in Chapter 3. Nonlinear time history analyses

are performed by using SAP2000 [21] for the scaled ground motion and maximum absolute

values of response parameters such as story displacements, inter-story drift ratios and story

shears are determined for each ground motion record. Plastic hinge locations are also

identified in nonlinear time history analyses.

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4.2 Time history analysis using selected earthquakes

Time history analysis means either an elastic or inelastic dynamic analysis of a structure

represented by a mathematical model through applying ground motion acceleration at its base

level. The time dependent dynamic response of the structure will be obtained through

numerical integration of its equation of motion. Time history analysis has been performed

with appropriate horizontal ground-motion time history components that has been selected

and scaled from different recorded events. Appropriate time histories have magnitude, fault

distance and source mechanism that are consistent with those that control the design basis

earthquake. Time history analysis can be of two types, namely elastic time history analysis

and nonlinear time history analysis. Nonlinear as well as linear time histories analyses have

been carried out and response of RC frames have been obtained for two appropriate ground

motions.

The behavior of RC under dynamic loading is not linear when the deformation is large and

load is time dependent. The use of linearly elastic analysis procedure is not sufficient to

capture the real behavior in such cases. In fact there are some situations where the use of such

simplified analyses can be misleading and missing in important details. The material and

geometric properties which are considered constant in linear analysis do not remain constant

in many practical situations. For example severe earthquake vibrations may cause quite large

structural deformations and as a result alter the stiffness properties significantly. Moreover,

member properties like mass or damping may undergo changes during the dynamic response,

while stiffness properties may vary significantly due to the material and geometric

nonlinearities caused by significant axial forces. Modal analysis is a commonly used method

of linear dynamic analysis, but is not valid for nonlinear systems. The incremental numerical

scheme needs to be applied for the dynamic analysis of nonlinear systems like reinforced

concrete structures.

During the last 150 years, seven major earthquakes (with M > 7.0) have affected the zone that

is now within the geographical borders of Bangladesh. Out of these, three had epicenters

within Bangladesh. However, well define peaks and segmented data not available for the

country. Earthquake data, especially earthquake induced forces are required for the purpose

of structural design, city planning and infrastructure development. From the geological study

it is evident that the intensity of earthquake hazard is not same throughout Bangladesh. So the

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whole country was divided in the national building code BNBC [1] into three seismic zones;

i.e., Zone1, 2 and 3.

Due to unavailability of seismic records, well-recorded available ground motions of El

Centro (Imperial Valley, 18 May 1940, NS component) and Kobe earthquake have been

selected for the current study. Figures 4.1 and 4.2 show the ground accelerations recorded

during some of the best known and widely studied earthquakes of the 20th century. Nonlinear

time history analysis of 2, 5 and 12-storied frames are performed by subjecting them to

recorded ground motions of El Centro (with ground acceleration ‘zone’ factor Z = 0.31) and

Kobe earthquake (Z = 0.55) using appropriate scaling.

4.2.1 El Centro 1940 Earthquake

The ground motion records used in this study include El Centro (Imperial Valley, 18 May

1940, NS component) earthquake. The N-S component (Peknold Version) recorded at a site

in El Centro, California during the Imperial Valley earthquake on 18 May 1940, has become

a popular point of reference for dynamic analyses (Han and Billington, 2004; Hueste and

Wight, 1999; Liang and Parra-Montesinos 2004; Madan et al. 1997; Chopra, 2003[5]; Ahmed,

1998; and so on). Many versions of this ground motion have been processed from the recorded

data. The version from Natural Center for Earthquake Engineering Research (NCEER) is

selected for this study. The peak ground accelerations (PGAs) of the selected ground motions

is 0.31882g and magnitude is 7.1. The duration of the acceleration time history utilized in this

study is 31.18s which is quite long, so provides clear picture of different responses and peaks

are also well defined and segmented, which is unique. Detail study of the frames responses

are carried out for El Centro 1940 earthquake and presented in this chapter.

Fig. 4.1 Ground acceleration of El Centro, 1940 earthquake record (N-S), adopted from

Chopra [5]

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4.2.2 Kobe Earthquake

Time acceleration data of the Kobe earthquake ground motions has also been used to analyze

RC plane frames for time history analysis. The peak ground acceleration (PGA) of the

selected ground motion is 0.55g. The duration of the acceleration time history utilized in this

study is 40.95sec. Figure 4.2 shows the ground acceleration from the Kobe (1995)

earthquake, which had caused major destructions in Japan, which is among the ‘best prepared

countries’ against earthquake disaster.

Fig. 4.2 Ground acceleration of Kobe Earthquake

4.3 Validation of linear time history analysis for SDOF system

In this chapter, the response of linear SDOF systems due to earthquake motion has been

studied. By definition linear systems are elastic systems and it is also referred as linearly

elastic systems to emphasize both properties.

4.3.1 Description of SDOF model SDOF models of target time periods have suitably been developed by selecting stiffness and

mass. To obtain a time period of T=1.0 sec, a 12 ft long vertical cantilever with a 4 inch

nominal diameter standard steel pipe and supporting a 5200 lbs weight attached at the tip is

chosen and is shown in Fig.4.3. The properties of the pipe are: Outside diameter= 4.50 inch,

Inside diameter = 4.026 inch, thickness = 0.237 inch and second moment of cross sectional

area, I = 7.23 in4, elastic modulus E = 29000 ksi. El Centro ground motion data has been

used in the analysis to determine the response of the SDOF system.

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Fig.4.3 SDOF Model

4.3.2 Analysis and Result

For a given ground motion (El Centro ground motion), the deformation response of an SDOF

system depends only on the natural vibration period of the system and its damping ratio.

Fig.4.4 shows the deformation response of three different systems due to El Centro ground

acceleration. The damping ratio, ξ=2% is the same for the three systems, so that only the

differences in their natural periods are responsible for the large differences in the deformation

responses. The damping ratio is the same for the three systems to the same ground motion. It

is observed the expected trend that systems with more period respond more deformation.

From Fig. 4.4(a) time period Tn=0.5 sec and ξ=2%, d = 2.67 inch, Fig. 4.4(b) time period Tn=

1 sec and ξ= 2%, d= 5.88 inch, Fig. 4.4(c) time period Tn= 2 sec and ξ= 2%, d= 7.53 inch.

Fig.4.5 shows these three systems, the longer the vibration period, the greater the peak

deformation. It is seen that the expected trend that systems with more damping respond less

deformation. From Fig. 4.5(a) time period Tn =2 sec and ξ=0%, d = 9.95 inch, Fig. 4.5(b)

time period Tn= 2 sec and ξ= 2%, d= 7.53 inch, Fig. 4.5(c) time period Tn= 2 sec and ξ= 5%,

d= 5.47 inch. The comparison of the peak deformations are shown in Table 4.1

Table 4.1 Comparison time history analyses with ETABS of SDOF system and published result from Chopra

As per Chopra As per Analysis Tn (sec) Damping ratio Displacement(in) Tn (sec) Damping ratio Displacement(in)

0.50 0.02 2.67 0.50 0.02 2.697501.00 0.02 5.97 1.00 0.02 5.882902.00 0.02 7.47 2.00 0.02 7.534982.00 0.00 9.91 2.00 0.00 9.952992.00 0.02 7.47 2.00 0.02 7.534982.00 0.05 5.37 2.00 0.05 5.47091

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(a) Tn= 0.5 sec, ξ=2%

(b) Tn= 1 sec, ξ=2%

(c) Tn= 2 sec, ξ=2% Fig. 4.4 Time history analysis of SDOF system for El Centro ground motion

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(a) Tn= 2 sec, ξ=0%

(b) Tn= 2 sec, ξ=2%

( c ) Tn= 2 sec, ξ=5%

Fig. 4.5 Time history analysis of SDOF system to El Centro ground motion

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Based on the above presented results, it is concluded that deformation responses of the

modeled of SDOF system under El Centro ground motion are very close to the published

responses of Chopra [5] and thus confirms the validity of the linear time history analysis.

4.4 Nonlinear Time History Analysis

Time history analysis means an elastic or inelastic dynamic analysis of a structure

represented by a mathematical model through applying a ground motion time history at its

base or any other appropriate level. The time dependent dynamic response of the structure

shall be obtained through numerical integration of its equation of motion. Time history

analysis shall be performed with appropriate horizontal ground motion time history

components that shall be selected and scaled from two recorded events. Appropriate time

histories shall have magnitude, fault distance and source mechanism that are consistent with

those that control the design basis earthquake (or maximum capable earthquake). The

parameters of interest shall be calculated for each time history analysis. If two time history

are performed, then the maximum response of the parameters of interest shall be used for

design. Time history analysis can be of two types, namely elastic time history analysis and

nonlinear time history analysis. Response parameters from elastic time history analysis shall

be denoted as elastic response parameters. All elements shall be designed using strength

design. Unlike static forces, the amplitude, direction and location of dynamic forces vary

significantly with time which causes considerable inertia effects on structures. Whereas the

static behavior of structures is solely dependent upon its stiffness characteristics, the behavior

of structures under dynamic forces is controlled by their mass, stiffness and damping

properties. Due to the large amplitudes caused by dynamic forces, performance of structures

depends on both the strength and deformability of constituent members, which is further

linked to their internal design forces, the prediction of which in turn depend upon the

accuracy of the method employed in their analytical determination. As linear dynamic

analysis procedures were developed, the code provisions were found to be inadequate in

providing the required structural strength of the building to withstand an intense earthquake.

Actual forces calculated by elastic analysis during earthquakes are much higher than the

design forces specified in the code. Moreover the behavior of construction materials like

reinforced concrete (RC) under dynamic loading is not linear; i.e., stress is not directly

proportional to strain which is valid for small deformation and load. In the case of earthquake

that produces intense load and large deformations the material properties vary with time and

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intensity of load. Hence the time varying and nonlinear characteristics of RC should be well

considered in structural analysis. Elastic response parameters may be scaled by using

appropriate response modification factor, R. Nonlinear time histories shall developed and

results determined for two appropriate ground motions. The maximum inelastic response

displacement shall not be reduced and shall comply with story drift limitations.

4.4.1 Validation of Nonlinear Time History Analysis for SDOF system with SeismoStruct

As mentioned earlier the behavior of RC under dynamic loading is not linear when the damage

stresses are greater than elastic limit. The use of linearly elastic analysis procedure is not valid

in such cases. In fact there are some situations where the use of such simplified analyses can be

misleading and missing in important details. The material and geometric properties which are

considered constant in linear analysis do not remain constant in many practical situations. For

example severe earthquake vibrations may cause quite large structural deformations and as a

result alter the stiffness properties significantly. Moreover, member properties like mass or

damping may undergo changes during the dynamic response, while stiffness properties may

vary significantly due to the material and geometric nonlinearities caused by significant axial

forces. The incremental numerical scheme needs to be applied for the dynamic analysis of

nonlinear systems like reinforced concrete structures. Nonlinear time history analysis has been

carried out using SeismoStruct [50] to predict the response of a full scale blind test of a bridge

pier [52] under six levels of ground motion. Detailed information about geometry material

properties and input files have been adopted from PEER website [52] and Bianchi [51].

SeismoStruct simulates the response quite reasonably.

4.4.1.1 Description of model

The model of this validation of nonlinear time history is a full scale reinforced concrete

bridge pier. The specimen was tested on the NEES Large High Performance Outdoor Shake

Table at UCSD’s Englekirk Structural Engineering Center under dynamic conditions, as part

of a blind prediction contest. Six uniaxial earthquake ground motions as shown in Figs 4.6,

4.7, 4.8, 4.9, 4.10 and 4.11 starting with low intensity shaking are increased so as to bring the

pier progressively to near collapse conditions. SeismoStruct is used in this analysis.

SeismoStruct[50] is a Finite Element package for structural analysis, capable of predicting

the large displacement behavior of space frames under static or dynamic loadings, taking into

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account both geometric nonlinearities and material inelasticity. The analytical results

obtained with SeismoStruct [50] in terms of displacements at the top of the column and base

shear are herein compared Figs.4.14 to 4.16 with the experimental observations. FE model is

defined in X-Z plane.

Fig. 4.6 Input Ground motion (EQ1)



85




86

4.4.1.2 Structural geometry and properties The model consists of a 1.22 m diameter cantilevered RC circular column with the properties

described hereafter. It is fixed at its support. The height and diameter of column is shown in

Fig. 4.12.

Fig. 4.12 Pier cross section and bridge pier specimen configuration

The concrete model is employed for defining the concrete materials. The characteristic

parameters are as follows:

For confined concrete: fc’= 41500 kPa, ft = 0, ξc = 0.0028 m/m, kc = 1.2, γ = 23.6 kN/m3

For unconfined concrete: fc’= 41500 kPa, ft = 0, ξc = 0.0028 m/m, kc = 1.0, γ = 23.6 kN/m3

For steel: Es = 2.000E+008 kPa, fy’= 518500 kPa, µ = 0.008, γ = 77 kN/m3

4.4.1.3 Modeling and loading

The column is modeled through a 3D force based inelastic frame element, where the number

of fibers used in section equilibrium computations is set to 300. Regarding the applied

masses, the effective mass of the pier is considered by assigning the specific weight of the

87

materials in the ‘materials’ module, whereas a lumped mass of 228 ton is concentrated at the

top of the pier.

In order to run a nonlinear dynamic analysis, a time history curve, constituted by six records

in series and separated by 10 seconds intervals with no acceleration, is loaded in the “Time

history Curve” dialog box. The applied ground motions are shown in Figs. 4.6 to 4.11. The

time step for the dynamic analysis is set as 0.00390625 sec. A 1% tangent stiffness

proportional damping is applied as global damping in this analysis.

A sketch of the FE model is presented in the following Fig. 4.13.

Fig. 4.13 FE model of the bridge column

4.4.1.4 Analysis type

Dynamic analysis is commonly used to predict the nonlinear inelastic response of a structure

subjected to earthquake loading (evidently, linear elastic dynamic response can also be

modeled for as long as elastic Elements and/or low levels of input excitation are considered).

The direct integration of the equations of motion is accomplished using the numerically

dissipative α-integration algorithm or a special case of the former, the well known Newmark

scheme [39], with automatic time step adjustment for optimum accuracy and efficiency (see

Automatic adjustment of load increment or time step). Modeling of seismic action is achieved

by introducing acceleration loading curves (accelerograms) at the supports, noting that

88

different curves can be introduced at each support, thus allowing for representation of

asynchronous ground excitation. In addition, dynamic analysis may also be employed for

modeling of pulse loading cases, in which case instead of acceleration time histories at the

supports, force pulse functions of any given shape (rectangular, triangular, parabolic, and so

on), can be employed to describe the transient loading applied to the appropriate nodes.

Nonlinear dynamic time history analysis is used in this chapter.

Several types of output can be obtained from the nonlinear dynamic time history analysis:

(a) Time versus displacement can be plotted, (b) Time versus base shear can be plotted and

(c) Time versus moment can be plotted.

Fig. 4.14 Analytical results at top displacement with time (EQ1 to EQ6)

Fig. 4.15 Analytical results at base shear with time (EQ1 to EQ6)

89

Fig. 4.16 Analytical results at moment with time (EQ1 to EQ6)

4.4.1.5 Comparison of analysis results The results of the analyses for the EQ1 earthquakes and experimental results are

superimposed on one plot of the roof displacement versus time and base shear versus time

shown in Figs 4.17, 4.18, 4.19, 4.20, 4.21 and 4.22. These figures demonstrates that the two

responses are almost same for most of the time history. The differences displayed in the plot

have been attributed to errors caused by different approaches to modeling the column

elements. These figures also demonstrates that the analytical displacement responses are very

similar to those obtains from experiment. The minor differences between analytical and

experimental can be attributed to assumption and simplification of modeling. For the

earthquake EQ1 gives a maximum analytical displacement at top of pier is 61.82mm and

maximum experimental displacement at top of pier is 62.98mm as shown in Fig.4.17 which is

close to same..

Fig. 4.17 Experiment vs. Analytical results at top displacement time (EQ1)

90

For the earthquake EQ3 gives a maximum analytical displacement at top level of pier is

185.44mm and maximum experimental displacement at top level of pier is 227.14mm as

shown in Fig.4.18 which is close to same..


For the earthquake EQ5 gives a maximum analytical displacement at top level of pier is

495.00mm and maximum experimental displacement at top of pier is 568.97mm as shown in

Fig.4.19 which is close to same but last displacements are far differences..


For the earthquake EQ1 gives a maximum analytical base shear at base level of pier is

511.34kN and maximum experimental base shear at base level of pier is 499.55kN as shown

in Fig.4.20 which is close to same..

91

Fig. 4.20 Experiment vs. Analytical results at base shear time (EQ1)



in Fig.4.21 which is close to same..




in Fig.4.22 which is close to same.

92


This chapter has done nonlinear dynamic time history analysis of the structure with

SeismoStructure [50] software. Base shear and displacements for dynamic load are extracted

from the database. Capacity curves (base shear versus roof displacement) are the load-

displacement envelopes of the structures and represent the global response of the structures.

Time history curves for case study bridge pier is obtained from the dynamic analyses using

dynamic load are almost same for blind test. The SDOF system modeled and analyses with

SeismoStruct [50] for the blind test pier is capable of simulating the dynamic responses

(displacement and base shear) of the actual experiment quite reasonably.

The Validation of the nonlinear dynamic time history analysis for SDOF system using

SeismoStruct with experimental results from the blind test [51] is close to same. Therefore, it

is concluded that nonlinear dynamic time history analysis run in SeismoStruct is valid and

accurate.

From above discussion, different figures we concluded that validation of nonlinear dynamic

time history analysis for SDOF system using SeismoStruct [50] is almost same to those of

results of blind test results [51].

4.5 Nonlinear time history analysis for 2D frame using SAP2000

The objective of this study is to verify that SAP2000, a relatively new computer program.

SAP2000 compared to SeismoStruct to prove the validity of a nonlinear time history analysis.

This comparison proved that when the different parameters are chosen carefully, the results

93

would match for the two programs. SeismoStruct is compared to SAP2000 in order to check

the validity of nonlinear pushover analysis. The results of these comparisons showed that

nonlinear analysis run in SAP2000 is valid and accurate.

The analysis of the case study model is performed by SAP2000. The results including the

displacement, forces and support reactions of the model are recorded. The total number

of mode shape used is equal to number of story. In mode shape lateral sway is considered

and the time period corresponding to that mode shape is taken.

Frame nonlinear hinge property is used to define nonlinear force displacement and/or moment

rotation behavior that can be assigned to discrete locations along the length of frame (line)

elements. These nonlinear hinges are only used during static nonlinear (pushover) analysis.

The built-in default hinge properties for concrete members are generally based on ATC-40.

Default hinge properties cannot be modified. They also can not be viewed because the default

properties are section dependent. The default properties cannot be fully defined by the

program until the section to which they apply is identified. Moment (M3) and shear (V2)

hinges is considered at each end of each beam and moment and axial force (P-M-M) is

considered at each end of column elements.

4.5.1 Analysis Technique

The step-by-step procedure is a second general approach to dynamic response analysis, and it

is well suited to analysis of nonlinear response because it avoids any use of superposition.

There are many different step-by-step methods, but in all of them the loading and the

response history are divided into a sequence of time intervals or “steps”. The response during

each step then is calculated from the initial conditions (displacement and velocity) existing at

the beginning of the step and from the history of loading during the step. Thus the response

each step is an independent analysis problem, and there is no need to combine response

contributions within the step. Nonlinear behavior may be considered easily by this approach

merely by assuming that the structural properties remain constant during each step and

causing them to change in accordance with any specified form of behavior from one step to

the next; hence the nonlinear analysis actually is a sequence of linear analyses of a changing

system. Any desired degree of refinement in the nonlinear behavior may be achieved in this

procedure by making the time steps short enough; also it can be applied to any type of

94

nonlinearity, including changes of mass and damping properties as well as the more common

nonlinearities due to changes of stiffness.

Step-by-step methods provide the only completely general approach to analysis of nonlinear

response; however, the methods are equally valuable in the analysis of linear response

because the same algorithms can be applied regardless of whether the structure is behaving

linearly or not. Moreover, the procedures used in solving single-degree-of-freedom structures

can easily be extended to deal with multi degree system merely replacing scalar quantities of

matrices. In fact, these methods are so effective and convenient that time-domain analyses

always are done by some form of step-by-step analysis regardless of whether or not the

response behavior is linear; the Duhamel method seldom is used in practice.

The response analysis procedures formulated in the time domain or in the frequency domain

involve evaluation of many independent response contributions that are combined to obtain

the total response. In the time domain procedure (Duhamel integral), the loading p(t) is

considered to be a succession of short-duration impulses, and the free-vibration response to

each impulse becomes a separate contribution to the total response at any subsequent time. In

the frequency-domain method, it is assumed that the loading p(t) is periodic and has been

resolved into its discrete harmonic components Pn by Fourier transformation. The

corresponding harmonic response components of the structure Vn are then obtained by

multiplying these loading components by the frequency response coefficient of the structure

Hn, and finally the total response of the structure is obtained by combining the harmonic

response components (inverse Fourier transform). Because superposition is applied to obtain

the final result in the both procedures, neither of these methods is suited for use in analysis of

nonlinear response; therefore judgment must be used in applying them in earthquake

engineering where it is expected that a severe earthquake will induce inelastic deformation in

a code-designed structure.

4.5.2 Analysis and results

The verification of perform is to validate the accuracy of the nonlinear inelastic time history

analysis. The 5-story structure is subjected to the El Centro and Kobe earthquake time history

that is previously used and is illustrated in Fig.4.23 and 4.24. The peak acceleration of the

loading history is scaled to 0.31g and 0.49g in order to obtain an inelastic response from the

structure. It is model inherent natural damping exactly the same in both programs; therefore it

95

is set to 1%. P-delta effect is also included in this analysis according to the approach

described in chapter 3.

The results of the analyses for the two programs are superimposed on one plot of the roof

displacement versus time shown in Figs. 4.23 and 4.24. For the El Centro earthquake gives a

maximum analytical displacement at top level of 5-story 2D frame is 57.46mm for SAP2000

and 63.31mm for SeismoStruct as shown in Fig.4.23 which is close to same. For the Kobe

earthquake gives a maximum analytical displacement at top level of 5-story 2D frame is

82.48mm for SAP2000 and 84.40mm for SeismoStruct as shown in Fig.4.24 which is close to

same. These figures demonstrate that the two responses are almost identical for most of the

time history. The differences displayed in the plot have been attributed to errors caused by

different approaches to modeling the beam and column elements. Therefore, it is concluded

that nonlinear dynamic time history analysis run in SAP2000 and SeismoStruct is valid and

accurate.

Fig. 4.23 Time History Analysis for 5-Story 2D Frame (El Centro earthquake)

Fig. 4.24 Time History Analysis for 5-Story 2D Frame (Kobe earthquake)

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4.6 Comparison of Linear and Nonlinear Time History Analysis

The linear and nonlinear time history analysis is used to estimate maximum seismic

displacement demands of 2, 5 and 12-story reinforced concrete frames under El Centro and

Kobe earthquakes. The displacements are compared obtained from linear and nonlinear time

history analysis and shown in Figs. 4.25, 4.26 and 4.27. SAP2000 [21] is used to perform

linear and nonlinear time history analyses. Nonlinear time history analyses of 2D frame

structures to determine displacements are performed by SAP2000 [21]. Target displacements

estimated using each procedure, values determined from nonlinear time history analyses for

all frames and ground motions are given in figures. The comparison of displacements

obtained from linear and nonlinear time history analyses could not reveal a clear particular

trend because structural response is affected by the variations in ground motion

characteristics and structural properties that each frame under each ground motion should be

considered as a case. However, the overall interpretation of results shows that the estimation

of approximate procedures yield different target displacement values than the exact results for

almost all cases. The accuracy of the predictions depends on ground motion characteristics

and structural properties as well as the inherent limitations of the procedures.

For the two story frame, the linear time history analysis gives a maximum dynamic

displacement at the roof of 70.48mm (2.56sec), minimum displacement 77.26mm (2.32sec)

and nonlinear time history analysis gives maximum dynamic displacement at the roof of

49.69mm (4.7sec), minimum displacement 50.65mm (4.9sec) as shown in Fig. 4.25.

Fig. 4.25 Comparison of linear and nonlinear time history analysis for 2-Story frame (El

Centro Earthquake)

97

For Kobe earthquake, the two story frame, the linear time history analysis gives a maximum

dynamic displacement at the roof of 70.48mm (12sec), minimum displacement 77.67mm

(11.74sec) and nonlinear time history analysis gives maximum dynamic displacement at the

roof of 45.31mm (11.52sec), minimum displacement 90.91mm (13.42sec) as shown in

Fig.4.26. From this figure it is seen that deformation is around 50 mm.

Fig. 4.26 Comparison of linear and nonlinear time history analysis for 2-Story frame

(Kobe Earthquake)

For the five story frame, the linear time history analysis gives a maximum dynamic

displacement at the roof of 146.87mm (5.82sec), minimum displacement 124.74mm


roof of 99.65mm (5.84sec), minimum displacement 103.62mm (5.46sec) as shown in Fig.

4.27.

Fig. 4.27 Comparison of linear and nonlinear time history analysis for 5-Story frame (El

Centro Earthquake)

98

For Kobe earthquake, the 5-story frame, the linear time history analysis gives a maximum

dynamic displacement at the roof of 217.48mm (14sec), minimum displacement 232.25mm



4.28.


(Kobe Earthquake)

For the twelve story frame, the linear time history analysis gives a maximum dynamic

displacement at the roof of 137.65mm (6.12sec), minimum displacement 117.88mm


roof of 84.16(6.12sec), minimum displacement 146.88mm (5.4sec) as shown in Fig. 4.29.


(El Centro Earthquake)

99

For Kobe earthquake, the 12-story frame, the linear time history analysis gives a maximum

dynamic displacement at the roof of 498.98mm (10.8sec), minimum displacement 481.56mm



4.30. From this figure it is seen that deformation is around 200 mm.


(Kobe Earthquake)

4.7 Comparison of capacity curves of pushover and nonlinear time history analysis

Most of the simplified nonlinear analysis procedures utilized for seismic performance

evaluation make use of pushover analysis and/or equivalent SDOF representation of actual

structure. However, pushover analysis involves certain approximations that the reliability and

the accuracy of the procedure should be identified. For this purpose, researchers investigated

various aspects of pushover analysis to identify the limitations and weaknesses of the

procedure and proposed improved pushover procedures that consider the effects of loading

patterns, higher modes, failure mechanisms, etc.

The nonlinear response of structures is very sensitive to the structural modeling and ground

motion characteristics. Therefore, different ground motion records that accounts for

uncertainties and differences in severity, frequency and duration characteristics has to be used

to predict the possible deformation modes of the structures for seismic performance

evaluation purposes. However, for simplicity, seismic demand prediction is generally

performed by pushover analysis which mostly utilizes smoothened response spectra. In this

100

study, the accuracy of demand prediction of pushover analyses for various loading patterns is

evaluated for the response obtained from selected ground motion data. In this study, the

response of case study frames are studied in the elastic and inelastic deformation levels that

are represented by peak roof displacements on the capacity (pushover) curve of the frames.

For each frame and each ground motion record is scaled to obtain the predetermined peak

roof displacement for the frame considered. The ground motion scale factors used to obtain

the predetermined peak roof displacements corresponding to the considered deformation

levels and the predetermined peak roof displacements are presented for reinforced concrete

frames in chapter 3. Nonlinear time history analyses are performed by using SAP2000 [21]

for the scaled ground motion records and maximum absolute values of response parameters

such as story displacements, inter-story drift ratios and story shears are determined at the

considered deformation for each ground motion record. It is also mentioning that the

maximum values of any response parameter over the height of the frames generally occurred

at different instants of time. Also, plastic hinge locations are identified in nonlinear time

history analyses. In pushover analyses, five different loading patterns are used in this study.

4.7.1 Description of case studies model

The pushover and nonlinear dynamic analysis are performed on moment resisting frames of

two displacement obtained from the pushover analysis can then be compared with the

maximum and minimum roof displacements induced by the dynamic analyses. The

dimensions of each frame are shown in Figs. 3.4, 3.5, 3.6 and also the fundamental, second,

third periods of each frame are shown in the Tables 3.2, 3.4 and 3.6.

The building frames are located in Dhaka area, on stiff soil, 5% structural damping, and were

designed for an earthquake with a 10% probability 50 years (Life safety). For the dynamic

load, the 1940 EI Centro ground motions are considered. Since a comparison is to be made

between dynamic and pushover responses, the ground motion should represent the same

conditions as the response spectrum. The 1940 EI Centro ground motions are selected for the

analyses.

4.7.2 Analysis and Result

The effects and the accuracy of invariant lateral load patterns utilized in pushover analysis to

predict the behavior imposed on the structure due to selected individual ground motions

101

causing elastic and various levels of nonlinear response are evaluated in this study. For this

purpose, global structure behavior, story displacements, inter story drift ratios, story shears

and plastic hinge locations are selected as response parameters. Pushover curves are obtained

by performing pushover analyses using SAP2000[21], story displacements, inter-story drift

ratios, story pushover curves and plastic hinge locations for any lateral load pattern are

extracted from the pushover analysis at the predetermined maximum roof displacement

consistent with the different level considered and are compared with absolute maximum

values of response parameters obtained from nonlinear time history analyses for each scale

factor for each ground motion. It should be mentioned that maximum story displacements and

inter story drift ratios for any story level generally occurred at different times in nonlinear

time history analyses. Also, story displacements; inter story drift ratios and plastic hinge

locations are estimated by performing pushover analysis on case study frames. Story

displacement, inter story drift ratio and corresponding error profiles for case study frames for

each pushover method at each scale factor for all ground motions are illustrated in this

chapter.

Location of weak points and potential failure modes that structure shall experience in case of

a seismic event is expected to be identified by pushover analyses. The accuracy of various

lateral load patterns utilized in traditional pushover analyses to predict the plastic hinges

similar to those predicted by nonlinear time history analyses is evaluated in this study. The

location of plastic hinges for case study R/C frames are predicted by pushover analyses

performed considering the lateral load patterns used in this study at roof displacements

corresponding to the nonlinear time history analysis by different scale factor. These

deformation levels represent low levels of nonlinear behavior, global yield and high levels of

nonlinear behavior. The pushover and nonlinear time history hinge patterns are compared.

The plastic hinge locations are also estimated for R/C frames by modal pushover analysis.

The ground motion scale factors used to obtain the peak roof displacements. Nonlinear time

history analysis are performed by SAP2000[21] for the scale ground motion records and

maximum absolute values of response parameters such as story displacements and base shear

are determined for El Centro ground motion record. It is also mentioning that the maximum

values of any response over the height of the frames generally occurred at different instance

of time. Capacity curve for 2-Story, 5-story and 12-Story frames are shown in Figs.4.31, 4.32

and 4.33. The base shear/weight (%) and story displacement/ height (%) data extracted from

thesis paper of Ogus [13] which are superimposed in Figs. 4.31, 4.32 and 4.33. From

102

Fig.4.31, 4.32 and 4.33, we have seen that capacity curves are close to same with different

loading pattern and nonlinear time history curve. Nonlinear time history analysis are

performed by SeismoStruct[50] for the scale ground motion records and maximum absolute

values of response parameters such as story displacements and base shear are determined for

El Centro ground motion record. It is also mentioning that the maximum values of any

response over the height of the frames generally occurred at different instance of time.

Capacity curve for 2-Story, 5-story and 12-story frames are shown in Figs.4.31, 4.32 and 4.33

which are close to SAP2000.

Fig. 4.31 Capacity Curve for 2-Story Frame (El Centro earthquake)



103

4.8 Performance Evaluation of the Structure using time history

The observations have been made from the comparison of plastic hinge locations determined

by pushover analyses and nonlinear time history analyses are locations of plastic hinges

obtained from nonlinear time history analyses are generally different for each ground motion

for each frame. None of the lateral load patterns could capture adequately the exact plastic

hinge locations obtained from nonlinear time history analyses at any considered deformation

level. Pushover analyses could not predict the plastic hinging in same sequence with nonlinear

time history analyses predictions. In this section two reinforced concrete frames 2 and 5 storied

2D frames are modeled. These frames are designed as per the provisions of BNBC [1].

Considering live load 29.19kN/m2 and dead load 58.38kN/m2 is used floor level. Self weight

of the concrete members considering unit weight of concrete as 23.56kN/m3, As per BNBC

seismic modification factor R=8 (IMRF) has been considered.

Performance point of any structure demand curve is required and demand curve can be

generated with SAP2000 [21]. But several parameters are required to generate the curves. In

this section, those parameters are defined and demand curve is plotted by SAP2000 [21]. The

performance point is determined for serviceability earthquake (SE), design earthquake (DE)

and maximum earthquake (ME).

4.8.1 Local level performance

Hinge curve for 2-Story frame for El Centro earthquake are shown in Fig.4.34 and for Kobe

earthquake are shown in Fig. 4.35. From Fig. 4.34 at performance point there is no hinge

form for ME level. From Fig. 4.35 at performance point there are 2 hinges for ME level

which are in the range of B-IO. There is no hinge form for SE and DE level. For three level

of earthquake, no hinge is found to cross the Immediate Occupancy (IO) limits. According

ATC-40[3], 2-story frame satisfies local criteria. Therefore it has been said that 2-story frame

structure fulfills the performance at local level for both earthquake.

104

Fig. 4.34 Deformation of the 2-Story 2D frame for ME (El Centro earthquake)

Fig. 4.35 Deformation of the 2-Story 2D frame for ME (Kobe earthquake)

Hinge curve for 5-Story frame for El Centro earthquake is shown in Fig.4.36 and for Kobe

earthquake is shown in Fig. 4.37. From Fig. 4.36 at performance point there are 10 hinges

which are in the range of B- IO. There is no hinge form for SE level. For three level of

earthquake, no hinge is found to cross the Immediate Occupancy (IO) limits. This 5-story

frame satisfies local criteria specified in ATC-40[3]. Therefore it has been said that 5-story

frame structure fulfills the performance at local level for El Centro earthquake and Kobe

earthquake.

105

Fig. 4.36 Deformation of the 5-Story 2D frame for ME (El Centro earthquake)

Fig. 4.37 Deformation of the 5-Story 2D frame for ME (Kobe earthquake)

4.8.2 Global level performance From the Figs. 4.38 and 4.39 it is seen that the performance point of 2-story frame structure

has maximum story drift of 0.0017 at story level 2 for El Centro earthquake and 0.0022 at

story level 2 for Kobe earthquake. These are less than allowable IO level 0.01 described in

ATC 40[3]. So this structure satisfies the requirement of ATC 40[3]. Therefore it has been

said that 2-story frame structure fulfills the performance objective at gloval level in

serviceability earthquake (SE), design earthquake(DE) and maximum earthquake(ME).

106

Fig. 4.38 Maximum story drift at performance point of 2-Story 2D frame for different earthquake level (El Centro earthquake)

Fig. 4.39 Maximum story drift at performance point of 2-Story 2D frame for different

earthquake level (Kobe earthquake)


From the Figs. 4.40 and 4.41 it is seen that the performance point of 5-story frame structure

has maximum story drift of 0.002024 at story level 2 for El Centro earthquake (ME) and

0.003006 at story level 2 for Kobe earthquake (ME). These are less than allowable IO level

0.01 described in ATC 40[3]. So this structure satisfies the requirement of ATC 40[3].

Therefore it has been said that 5-story frame structure fulfills the performance objective at

107

gloval level in serviceability earthquake (SE), design earthquake(DE) and maximum

earthquake(ME).

Fig. 4.40 Maximum story drift at performance point of 5-Story 2D frame for different

earthquake level (El Centro earthquake)

Fig. 4.41 Maximum story drift at performance point of 5-Story 2D frame for different earthquake level (Kobe earthquake)


From above discussion, different figures, it has been concluded that the performance point of

structure has maximum story drift ratio for 2-story frame and 5-story frames are less than IO

108

level in maximum earthquake level. These two frame satisfy local and global criteria

specified in ATC-40[3].. Therefore it has been said that both the structure fulfills the

performance objective at gloval level in serviceability earthquake (SE), design

earthquake(DE) and maximum earthquake(ME). .

4.9 Conclusion Pushover analysis is a simplified nonlinear analysis to develop capacity curves of a structure

and determine their performance. However, the correctness of using this analysis is

sometimes questionable; particularly the load pattern used in pushover analysis to push the

structure to failure is an important issue and should be able to model the real response under a

seismic load. On the other hand, a nonlinear time history analysis (NLTHA) is capable of

modeling the real seismic response under a real earthquake ground motion, although the

method is time consuming and complicated. An important part of this current work is to

compare capacity curves of pushover analysis with different load patterns and compare them

with NLTHA.

In this chapter, linear time history analysis (LTHA) has been validated against published

results of Chopra for a SDOF system. A nonlinear time history analysis (NLTHA) using

SeismoStruct has also been tried to simulate an experiment where a bridge pier had been

subjected to six different levels of ground motion and the results are comparable with the

experiment. NLTHA was also carried out using SAP2000 [21] and the results were found to

compare well with those of Seismostruct. A comparative study was done to observe the effect

of damage in a NLTHA if used instead of linear analysis. NLTHA were extensively carried

out on three frames to develop capacity curves and compare them with those obtained from

pushover analysis with different load pattern e.g. triangular, elastic first mode and uniform.

These analyses show that pushover analysis with uniform load pattern compares well with

that of NLTHA. The analysis also showed that current pushover curves were similar to some

published results. A performance evaluation of RC frames were carried out using NLTHA

and showed that frames designed by BNBC for Dhaka city can easily satisfy the local and

global performance requirements.

109

Chapter 5

Performance based analysis of RC frame building

5.1 Introduction

In Chapter 3, pushover analysis method has been validated by comparing capacity curves of

2D frames with published results. Demand curves are also checked by comparing procedures

A and B. Performances of 2D frames designed by BNBC [1] has been found to satisfy

required local and global performance levels. More rigorous nonlinear time history analysis

has been studied in Chapter 4 where it has been validated against published result.

Performance of 2D frames have also been checked by nonlinear time history analysis and

BNBC designed frames have been found to satisfy the requirements. Loading patterns used in

pushover analysis has been studied by comparing the capacity curves with those of nonlinear

time history analysis to determine the more suitable pattern. Nonlinear time history analysis

is more correct and rigorous method, however, use of nonlinear time history analysis is not

suitable for everyday design office use and pushover analysis method can be a simpler

alternative. In this chapter, pushover analysis is carried out on 3D buildings designed by

BNBC [1] to check their performance adequacies for local and global cases.

A six storied building with RC moment resisting frames has been analyzed and design as per

the provisions of BNBC [1] and pushover analysis is carried out to find the capacity curve.

Local and global performances are investigated for three levels of earthquake demand for

Dhaka city.

5.2 Performance requirements

To determine whether a building meets a specified performance objective, response quantities

from a nonlinear static are compared with limits for appropriate performance levels. This

chapter presents those structural response limits, which constitute acceptance criteria for the

building structure. The response limits falls into two categories:

a) Global acceptable limits: These response limits include requirements for the vertical

load capacity, lateral load resistance and lateral drift.

110

b) Element and component acceptability limits: Each element (frame, wall, diaphragm, or

foundation) must be checked to determine if its components respond within acceptable

limits.

Building performance objectives are checked against some predefined seismic demand.

Seismic demand for a structure is totally site depended. For analysis development of site

dependent elastic response spectrum is needed. The Federal Emergency Management Agency

(FEMA-356) has recommended standard procedure to establish seismic demand at a site.

5.3 Description of 6-story reinforced concrete frame structure building

The structure is a six story residential building. The building is a 4x3 bay immediate moment

resisting frame of grid 6m in both sides. It is fixed at its support at 2.0m below the existing

ground level. Typical floor height is 3.0m except 4.0m at the ground floor for parking. Other

structural dimensions are given in Table 5.1.

Fig. 5.1 Layout of the 6-story building

Table 5.1 Structural dimension of 6-story building

Element Above ground Below ground Clear cover to re-bar center

COLUMN Exterior 300x450mm 375x525mm 50.0 mm above ground and 75.0 mm below ground Interior 300x550mm 375x625mm

BEAM

Grade Beam 325x455mm 75.0 mm all sides

Floor Beam 250x450mm 50.0 mm all sides

SLAB All Floor 140mm

111

Default concrete material was used in the design of RC beam column having the following

property:

28 days Cylinder strength of Concrete, f’c = 25 MPa

Yield strength of Steel, fy = 410 MPa

Modulus of elasticity, Ec = 24821 Mpa

Standard steel bar is used as reinforcing material.

Load used in design:

The Self weight of the structure is calculated automatically by the program. In addition to floor

finish 1 kN/m2 distributed on the floors including roof, live load 2 kN/m2 are considered as

dead load in typical floor level including roof. Self weight of the concrete members considering

unit weight of concrete as 24 kN/m3, 250mm Brick work on the exterior beams and 125 mm

brick work on the interior beams. Though there are no wall in the parking floor (ground Floor),

125 mm B/W assumed on the grade beams. Parapet on the roof considered as 125 mm brick

work of height 1.50 m. Unit wt. of brick work considered as 23 kN/m3. Here site location is

Dhaka. As per BNBC [1] seismic modification factor R=8 (IMRF) has been considered. Other

co-efficient used as

Seismic zone coefficient, Z=0.15 for Zone 2, Structure importance coefficient, I= 1.00

Site coefficient, S=1.5

Assumption for pushover Analysis

ETABS is used here for the pushover analysis. The assumptions that are considered for the

analysis by this software are as follows:

1) Moment (M3) and shear (V2) hinge is considered at each end of each beam and moment and

axial (P-M-M) is considered at each end of column elements.

2) Three nonlinear cases are defined here. (a) Pushover analysis has been done using load

pattern of equivalent static earthquake load calculated by program.

(b) Dead load was considered as the previous pushover cases for each analysis.

(c) Unload entire structure is selected for distribution of loads when local hinges fail.

(d) Geometric non-linearity (P-∆ effect) is considered with full dead load.

(e) Unload entire structure is considered for member unloading method.

(f) Horizontal displacement of topmost corner node has been selected for performance

monitoring of roof displacement.

3) Special seismic design data are not included here.

112

4) In analysis option P-∆ effect and dynamic analysis option are considered.

5.3.1 Performance evaluation of a structure design as per BNBC

In Bangladesh, buildings are designed according to BNBC [1]. In this section, a structure,

designed for gravity load and earthquake loads are analyzed to assess its performance. The

geometry and other structural details are mention in the previous section 5.3. The design

followed as BNBC [1] including earthquake and gravity loads. Performance point of the

structure is evaluated for serviceability, design and maximum earthquake.

Performance point of any structure, demand curve is required and demand curve has been

generated with ETABS [20]. But several parameters are required to generate the curves. In

this section, those parameters are defined and demand curve is plotted by ETABS [20].

Capacity curves (base shear versus roof displacement) are the load-displacement envelopes of

the structures and represent the global response of the structures. The maximum values of

roof displacements and base shear are determined for deformation level to approximate a

dynamic capacity curve for the building frame. The performance point is determined and

compared for different level of earthquakes.

Establishment Demand Spectra: Location of the site: Dhaka city

Soil profile at the site: Soil type SC as per Table 2.5 and 2.6 when the soil properties are not

known in sufficient detail.

Table 5.2 Calculation of CA Seismic Zone Factor,Z 0.15 As per BNBC/93 0.15 As per BNBC/93 0.15 As per BNBC/93 Earthquake Hazard Level, E 0.5 Design Earthquake 1 Max Earthquake 0.35 Serviceability Earthquake Factored E (E'X1.4) 0.7 1.4 0.49

Near-Source Factor 1 >15km, table 2.4 1 >15km, table 2.4 1 >15km, table 2.4 Shaking Intensity, ZEN 0.105 0.21 0.0735 For Soil Type SC, CA 0.126 From Table 2.5 0.249 From Table 2.5 0.088 From Table 2.5 Table 5.3 Calculation of CV

Seismic Zone Factor,Z 0.15 as per BNBC/93 0.15 as per BNBC/93 0.15 as per BNBC/93 Earthquake Hazard Level, E 0.5 Design Earthquake 1 Max Earthquake 0.35 Serviceability Earthquake Factored E (E'X1.4) 0.7 1.4 0.49 Near-Source Factor 1 >15km, table 2.4 1 >15km, table 2.4 1 >15km, table 2.4 Shaking Intensity, ZEN 0.105 0.21 0.0735 For Soil Type SC, CV 0.178 From Table 2.6 0.333 From Table 2.6 0.128 From Table 2.6

113

Table 5.4: Calculation of reduction factor

Design Earthquake Max Earthquake Serviceability Earthquake

X-dir Y-dir X-dir Y-dir X-dir Y-dir

Effective damping, βeff

12.10 10.70 22.10 25.40 7.60 6.20

Spectral acceleration

reduction factor, SRA

0.71 0.75 0.52 0.48 0.86 0.93

Spectral velocity

reduction factor, SRV

0.78 0.81 0.63 0.60 0.90 0.95

Effective peak ground

acceleration (EPA)

0.126g 0.126g 0.249g 0.249g 0.088g 0.088g

Average value of peak

response

0.225g 0.237g 0.324g 0.297g 0.190g 0.204g

TA 0.123 sec 0.122 sec 0.129 sec 0.134 sec 0.121 sec 0.119 sec

Ts 0.617 sec 0.608 sec 0.647 sec 0.669 sec 0.604 sec 0.593 sec

The capacity curve is superimposed on the response spectrum curve in ADRS format. It is

seen from the analysis that the capacity curve intersects the demand curve. Performance point

is the intersection point.

Fig. 5.2 Capacity spectrum of the six-story frame structure building in x-direction

(Procedure A)

114

Fig. 5.3 Capacity spectrum of the six-story frame structure building in y-direction

(Procedure A)

Fig. 5.4 Capacity spectrum of the six-story frame structure building in x-direction (SE)

(Procedure B)

115

Fig. 5.5 Capacity spectrum of the six-story frame structure building in x-direction (DE)

(Procedure B)

Fig. 5.6 Capacity spectrum of the six-story frame structure building in x-direction (ME)

(Procedure B)

116

Fig. 5.7 Capacity spectrum of the six-story frame structure building in y-direction (SE)

(Procedure B)

Fig. 5.8 Capacity spectrum of the six-story frame structure building in y-direction (DE)

(Procedure B)

117

Fig. 5.9 Capacity spectrum of the six-story frame structure building in y-direction (ME)

(Procedure B)

From procedure A and B capacity curve illustrated in Figs 5.2 to 5.9. The performance point

spectral displacement and spectral acceleration data for different earthquake are in Table 5.5.

Table 5.5 Capacity spectrum of the six story building (Procedure A and B)

X-direction Y-direction

Spectral acceleration Spectral displacement, mm


Procedure A

Procedure B

Procedure A

Procedure B

Procedure A

Procedure B

Procedure A

Procedure B

Serviceability earthquake

0.085 0.088 38.00 37.22 0.075 0.070 51.00 52.80

Design earthquake

0.098 0.097 49.96 49.85 0.080 0.078 65.51 66.61

Maximum earthquake

0.126 0.116 78.63 94.24 0.081 0.079 118.20 124.24

From procedure A and B (Table 5.5) and Figs.5.4 to 5.9, it has seen that performance point

spectral displacement and spectral acceleration are close to same. So it has been concluded

that capacity spectrum of six story frame structure building is close to same for Procedure A

and B.

118


For serviceability earthquake, number of hinges formed in x and y direction does not cross

the Immediate Occupancy (IO) limit (Table 5.6). So it can be concluded that the local criteria

as per ATC-40 [3] is satisfied for serviceability earthquake.

For design earthquake, number of hinges formed in x and y direction does not cross the Life

Safety (LS) limit (Table 5.6). So it can be concluded that the local criteria as per ATC-40 [3]

is satisfied for design earthquake.

For maximum earthquake, number of hinges formed in x and y direction does not cross the

Collapse prevention (CP) limits (Table 5.6). So it can be concluded that the local criteria as

per ATC-40 [3] is satisfied for maximum earthquake.

Therefore it has been said that the structure fulfills the performance objective at local level

and performance of building frames designed as per BNBC has been evaluated against

targeted performance levels for serviceability, design and maximum earthquakes.

Table 5.6 Number of hinges formed in the 6-storied frame structure building in x and y-

direction

A-B B-IO IO-LS LS-CP CP-C C-D D-E >E TOTAL

X-direction

Serviceability earthquake 592 122 0 0 0 0 0 0 714

Design earthquake 558 96 60 0 0 0 0 0 714

Maximum earthquake 538 112 64 0 0 0 0 0 714

Y-direction





119

Fig. 5.10 Deformation of the building at performance point in x-direction for SE

Fig. 5.11 Deformation of the building at performance point in x-direction for DE

Fig. 5.12 Deformation of the building at performance point in x-direction for ME

120

Fig. 5.13 Deformation of the building at performance point in y-direction for SE

Fig. 5.14 Deformation of the building at performance point in y-direction DE


In this section, a structure, designed for gravity load and earthquake as per BNBC [1] is

analyzed to assess its performance. The geometry and other structural details are same as

mention in the previous section 5.3. The design followed as BNBC including earthquake and

gravity loads. Performance point of the structure is evaluated for serviceability, design and

maximum earthquake.

For serviceability earthquake, from Figs. 5.15 and 5.16, it is seen that story drift ratio does

cross the Immediate Occupancy (IO) limits. So it can be concluded that structure is designed

for earthquake and gravity load as per BNBC [1] satisfy at small earthquake.

121

For design earthquake, from Figs. 5.15 and 5.16, it is seen that story drift ratio does not cross

the Life Safety (LS) limits. So it can be concluded that the global criteria as per ATC-40 [3]

is satisfied for design earthquake.

For maximum earthquake, From Figs. 5.15 and 5.16, it is seen that story drift ratio does not

cross the Life Safety (LS) limits. So it can be concluded that the global criteria as per ATC-

40 [3] is satisfied for maximum earthquake.

Therefore it has been said that the structure fulfills the performance objective at global level



Fig. 5.15 Maximum story drift ratio at performance point for different earthquake level

in X-direction


in Y-direction


122

5.4 Performance of 6-story RC narrow building

In section 5.3 a 4X3 bay building, designed as per BNBC, has been analyzed to check its

performance. In this current section, a somewhat narrow building is considered to observe

whether it can satisfy the performance requirements as before. The geometry and other

structural details are mentioned in the previous section 5.3. The building is a 4x2 bay

immediate moment resisting frame of grid 6m in X-direction and 6m in Y-direction. It is

fixed at its support at 2.0m below the existing ground level. Typical floor height is 3.0m

except 4.0m at the ground floor for parking. Other structural dimensions are given in Table

5.7.

Fig. 5.17 Layout of the 6-story building




BEAM



SLAB All Floor 140mm Materials properties are same as mention in the previous section 5.3. Load used in design are

same as mention in the previous section 5.3.

123


















Table 5.8 Calculation of reduction factor Design Earthquake Max Earthquake Serviceability Earthquake



13.10 10.50 22.10 23.50 8.10 6.50

Spectral acceleration

reduction factor, SRA

0.69 0.76 0.52 0.50 0.84 0.91

Spectral velocity

reduction factor, SRV

0.76 0.82 0.63 0.62 0.88 0.93

Effective peak ground

acceleration (EPA)

0.126g 0.126g 0.249g 0.249g 0.088g 0.088g

Average value of peak

response

0.217g 0.230g 0.324g 0.312g 0.185g 0.201g



124




Fig. 5.18 Capacity spectrum of the six-story frame structure building in x-direction

(Procedure A)

Fig. 5.19 Capacity spectrum of the six-story frame structure building in y-direction

(Procedure A)

125

Fig. 5.20 Capacity spectrum of the six-story frame structure building in x-direction (SE)

(Procedure B)

Fig. 5.21 Capacity spectrum of the six-story frame structure building in x-direction (DE)

(Procedure B)

126

Fig. 5.22 Capacity spectrum of the six-story frame structure building in x-direction (ME)

(Procedure B)

Fig. 5.23 Capacity spectrum of the six-story frame structure building in y-direction (SE)

(Procedure B)

127

Fig. 5.24 Capacity spectrum of the six-story frame structure building in y-direction (DE)

(Procedure B)

Fig. 5.25 Capacity spectrum of the six-story frame structure building in y-direction (ME)

(Procedure B)

128



Table 5.9 Capacity spectrum of the six story building (Procedure A and B) X-direction Y-direction

Spectral acceleration Spectral displacement,

mm

Spectral acceleration Spectral displacement,

mm

Procedure

A

Procedure

B

Procedure

A

Procedure

B

Procedure

A

Procedure

B

Procedure

A

Procedure

B

Serviceability

earthquake

0.097 0.095 34.54 33.42 0.081 0.073 45.17 49.48

Design

earthquake

0.106 0.105 45.58 43.65 0.089 0.083 64.88 63.55

Maximum

earthquake

0.125 0.126 84.32 86.31 0.09 0.09 125.44 115.48

From procedure A and B (Table 5.9) and Figs.5.20 to 5.25, it has seen that performance point

spectral displacement and spectral acceleration are close to same. So it has been concluded

that capacity spectrum of six story frame structure building is close to same for Procedure A

and B.



the Immediate Occupancy (IO) limit (Table 5.10). For design earthquake, number of hinges

formed in x and y direction does not cross the Life Safety (LS) limit (Table 5.10). For

maximum earthquake, number of hinges formed in x and y direction does not cross the


per ATC-40 [3] is satisfied for serviceability, design and maximum earthquake.




129

Table 5.10 Number of hinges formed in the 6-storied frame structure building in x and y-direction


X-direction




Y-direction










maximum earthquake.

For serviceability earthquake, from Figs. 5.26 and 5.27, it is seen that story drift ratio does

cross the Immediate Occupancy (IO) limits. For design earthquake, from Figs. 5.26 and 5.27,

it is seen that story drift ratio does not cross the Life Safety (LS) limits. For maximum

earthquake, from Figs. 5.26 and 5.27, it is seen that story drift ratio does not cross the Life

Safety (LS) limits. So it can be concluded that the global criteria as per ATC-40 [3] is

satisfied for serviceability, design and maximum earthquake.




130


in X-direction


in Y-direction


A typical 6-story building with aspect ratio of 0.5, situated in Dhaka city (Zone 2) was analyzed

using finite element methodology developed in this study. The results obtained from the

analysis are studied and seismic performances are checked against ATC-40[3] local and global

requirements. It has been seen that the story drifts for maximum, design, and serviceability

131

earthquakes are within specified limits. It has also been found that number of hinges formed in

the structure does not cross the specified limits for three levels of earthquakes.

5.5 Performance of 12-story RC frame building

In sections 5.3 and 5.4, 6-storied buildings have been considered to check their seismic

performances. In the current section, the structure chosen for analysis is a 12-story residential

building. The building is a 4x3 bay immediate moment resisting frame of grid 6m in both

sides. It is fixed at its support at 2.0m below the existing ground level. Typical floor height is

3.0m except 4.0m at the ground floor for parking. Other structural dimensions are given in

Table 5.11.

Fig. 5.28 Layout of 12-story building

132

Fig. 5.29 Elevation of 12-story building




BEAM



SLAB All Floor 140mm

Materials properties are same as mention in the previous section 5.3. Load used in design are

same as mention in the previous section 5.3.

133


















Table 5.12 Calculation of reduction factor

Design Earthquake Max Earthquake Serviceability Earthquake



11.80 10.70 22.90 24.70 6.70 5.90

Spectral acceleration reduction factor, SRA

0.72 0.75 0.51 0.49 0.90 0.94

Spectral velocity reduction factor, SRV

0.79 0.81 0.62 0.60 0.93 0.96

Effective peak ground acceleration (EPA)

0.126g 0.178g 0.249g 0.333g 0.088g 0.128g

Average value of peak response

0.228g 0.237g 0.317g 0.303g 0.199g 0.208g



134





(Procedure A)


(Procedure A)

135

Fig. 5.32 Capacity spectrum of the 12-story frame structure building in x-direction (SE)

(Procedure B)

Fig. 5.33 Capacity spectrum of the 12-story frame structure building in x-direction (DE)

(Procedure B)

136

Fig. 5.34 Capacity spectrum of the 12-story frame structure building in x-direction (ME)

(Procedure B)

Fig. 5.35 Capacity spectrum of the 12-story frame structure building in y-direction (SE)

(Procedure B)

137

Fig. 5.36 Capacity spectrum of the 12-story frame structure building in y-direction (DE)

(Procedure B)

Fig. 5.37 Capacity spectrum of the 12-story frame structure building in y-direction (ME)

(Procedure B)

138



Table 5.13 Capacity spectrum of the 12 story building (Procedure A and B)

X-direction Y-direction



Procedure A

Procedure B

Procedure A

Procedure B

Procedure A

Procedure B

Procedure A

Procedure B

Serviceability earthquake

0.061 0.059 62.30 59.79 0.054 0.051 79.22 74.64

Design earthquake

0.066 0.064 80.00 77.09 0.055 0.055 92.77 93.38

Maximum earthquake

0.069 0.073 148.00 145.48 0.067 0.055 150.00 179.70

From procedure A and B (Table 5.13) and Figs.5.30 to 5.37, it has seen that performance

point spectral displacement and spectral acceleration are close to same. So it has been

concluded that capacity spectrum of 12-story frame structure building is close to same for

Procedure A and B.



the Immediate Occupancy (IO) limit (Table 5.14). For design earthquake, number of hinges

formed in x and y direction does not cross the Life Safety (LS) limit (Table 5.14). For

maximum earthquake, number of hinges formed in x and y direction does not cross the


per ATC-40 [3] is satisfied for serviceability, design and maximum earthquake.




139

Table 5.14 Number of hinges formed in the 12-storied frame structure building in x and

y-direction


X-direction Serviceability earthquake 1174 152 0 0 0 0 0 0 1326Design earthquake 1116 146 64 0 0 0 0 0 1326Maximum earthquake 1058 140 128 0 0 0 0 0 1326Y-direction Serviceability earthquake 1236 90 0 0 0 0 0 0 1326Design earthquake 1208 118 0 0 0 0 0 0 1326Maximum earthquake 1144 72 30 80 0 0 0 0 1326







maximum earthquake.

For serviceability earthquake, From Figs. 5.38 and 5.39, it is seen that story drift ratio does

cross the Immediate Occupancy (IO) limits. So it has been said that structure is designed for

earthquake and gravity load as per BNBC [1] satisfy at small earthquake. For design

earthquake, From Figs. 5.38 and 5.39, it is seen that story drift ratio does not cross the Life

Safety (LS) limits. For maximum earthquake, From Figs. 5.38 and 5.39, it is seen that story

drift ratio does not cross the Life Safety (LS) limits. So it can be concluded that the global

criteria as per ATC-40 [3] is satisfied for serviceability, design and maximum earthquake.




140


in X-direction


in Y-direction


A typical 12-story building situated in Dhaka city (Zone 2) was analyzed using finite element

methodology developed in this study. The results obtained from the analysis are studied and

seismic performances are checked against ATC-40[3] local and global requirements. It has

been seen that the story drifts for maximum, design, and serviceability earthquakes are within

specified limits. It has also been found that number of hinges formed in the structure does not

cross the specified limits for three levels of earthquakes.

141

5.6 Performance evaluation of structure for gravity loads only

In this section, a structure, designed only for gravity load is analyzed to assess its

performance. The geometry and other structural details are same as mention in the previous

section 5.3. The design followed as BNBC [1] excluding earthquake and wind loads.

Performance point of the structure is evaluated for serviceability, design and maximum

earthquake.

For serviceability earthquake, number of hinges formed in x and y direction cross the

Immediate Occupancy (IO) limits (Table 5.15). From Figs. 5.40 and 5.41, it is seen that story

drift ratio cross the Immediate Occupancy (IO) limits. So it can be concluded that structure

designed only for gravity load as per BNBC [1] fails even at small earthquake.

For design earthquake, number of hinges formed in x and y direction does not cross the Life

Safety (LS) limits (Table 5.15). From Figs. 5.40 and 5.41, it is seen that story drift ratio does

not cross the Life Safety (LS) limits. So it can be concluded that the local and global criteria

as per ATC-40 [3] is satisfied for design earthquake.

For maximum earthquake, performance point for this structure is not found. From table 5.15,

number of hinges formed in x and y direction cross the Collapse prevention (CP) limits

(Table 5.15). So it can be concluded that structure designed only for gravity load as per

BNBC [1] fails at maximum earthquake level.

Table 5.15 Number of hinges for different level of earthquakes

A-B B-IO IO-LS LS-CP CP-C C-D D-E >E TOTALX-direction Serviceability earthquake 546 82 62 24 0 0 0 0 714Design earthquake 514 72 88 40 0 0 0 0 714Maximum earthquake* 512 38 52 108 0 4 0 0 714Y-direction Serviceability earthquake 579 50 60 25 0 0 0 0 714Design earthquake 576 48 62 28 0 0 0 0 714Maximum earthquake* 535 64 61 50 0 2 2 0 714


*Corresponding to last step of pushover curve

142


in x-direction


in y-direction


Performance point for ME has not been achieved

143

5.7 Conclusion

In this chapter, performance based analysis of different 3D RC buildings, designed as per

BNBC [1] have been carried out. Typical buildings situated in Dhaka city (Zone 2) with

different aspect ratio and story heights were analyzed using finite element methodology

developed in this study. The results obtained from the analyses are studied and seismic

performances are checked against ATC-40[3] local and global requirements. It has been seen

that the story drifts for maximum, design, and serviceability earthquakes are within specified

limits for all cases considered. It has also been found that number of hinges formed in the

structure does not cross the specified limits for three levels of earthquakes. The buildings

designed following seismic provisions of BNBC [1] would behave rather conservatively during

an earthquake.

It has been seen that spectral displacement and spectral acceleration of performance points are

almost same for procedure A and B. It can be concluded that demand requirements of 3D frame

structures are same for procedure A and B and either method can be used.

The chapter also shows the importance of considering seismic force in design. A building

designed only for gravity load ignoring the seismic forces has been found to fail in satisfying

the performance requirement levels particularly those for the serviceability and maximum

earthquakes.

144

Chapter 6

Conclusions and Recommendations

6.1 Introduction

In this thesis, an attempt has been made to demonstrate the validity and efficiency of

pushover analysis method of ATC-40[3] as incorporated in ETABS [20], SAP2000 [21] and

SeismoStruct [50]. The work also studied the effectiveness of pushover analysis in

comparison to more rigorous nonlinear time history analysis with particular emphasis on the

load pattern employed in pushover analysis. Finally performance of building frames designed

as per BNBC has been evaluated against targeted performance levels for serviceability,

design and maximum earthquakes.

In Chapter 2, provisions of BNBC [1] for seismic design have been discussed in detail.

Fundamentals of pushover analysis as proposed by ATC-40[3] have also been presented.

Pushover is a simplified nonlinear analysis to determine the performance of a structure for a

given level of earthquake. ATC-40[3] gives provisions to obtain capacity and demand curves;

also it specifies local and global requirements for different levels of earthquake. A brief

description of nonlinear time history analysis has been presented in Chapter 2. In Chapter 3,

Pushover analysis method as incorporated in ETABS [20] and SAP2000 [21] has been

validated by comparing the capacity curves of 2D frames with published results. Pushover

analyses have been carried out on 2D frames designed according to BNBC and their

performances have been found to be satisfactory as per the requirements of ATC-40[3].

Different load pattern have been used in the pushover analysis to demonstrate their effect on

capacity curve.

Chapter 4 presents linear and nonlinear time history analysis using SAP2000 [21] and

SeismoStruct [50]. Both linear and nonlinear time history analyses have been validated

against published results. Nonlinear time history analysis, although complicated and time

consuming, is a more rigorous method for modeling seismic response of a structure. Different

load patterns i.e. triangular, uniform, elastic first modes have been used in pushover analysis

to identify the most suitable one that compares well with nonlinear time history analysis.

Performances have been evaluated for 2D frames designed according to BNBC [1] using

nonlinear time history. Well known earthquake i.e. El Centro and Kobe earthquakes have

suitably scaled as per required Z value for use in the analysis. A comparison of linear and

145

nonlinear time history analyses have been carried out to demonstrate that damage caused

during an earthquake. In Chapter 5, similar pushover analyses have been carried out for 3D

buildings designed according to BNBC [1] to determine their performance under three levels

of earthquakes. A building which is designed only for gravity load has also been studied to

see whether it is capable of satisfying the requirements of ATC-40 [3].

6.2 Findings of the study

The findings of the study regarding the effectiveness of pushover and time history analyses as

implemented in finite element method can be summarized as follows:

1) Capacity curves of pushover analysis as obtained by ETABS and SAP2000 have been

validated with published numerical result of Oguz [13]. The curves obtained by

SAP2000 [21] and ETABS [20] are almost identical in elastic range and compares well

in the inelastic range.

2) Performance based analysis of the structure with SeismoStruct [50] software has also

been carried out. Capacity curve of pushover analysis using SeismoStruct has been

compared to that of SAP2000[21] and two software, even though different in their

formulation, yield comparable results.

3) Linear time history analysis of ETABS [20] has been validated for SDOF system by

comparing the response with published results of Chopra [5].

4) Nonlinear time history analysis has been carried out using SeismoStruct [50] to predict

the response of a full scale blind test of a bridge pier [52] under six levels of ground

motion. SeismoStruct simulates the response quite reasonably.

5) Effectiveness of pushover analysis depends significantly on the load pattern used in

finding the capacity curve. Different loading patterns have been compared with the

results of nonlinear time history analysis and uniform loading pattern have been found

to compare better.

6) A comparative study was done to observe the effect of damage under seismic loading

by analyzing using both nonlinear time history and linear time history analysis.

7) Two methods of ATC-40 [3], i.e. Procedure A and Procedure B, have been compared to

find the demand curve of pushover analysis. It has been seen that spectral displacement

and spectral acceleration corresponding to performance point are almost same for

procedure A and B.

146

Typical buildings situated in Dhaka city (Zone 2) with different aspect ratio and story

heights were analyzed using finite element methodology developed in this study. It has

been found that the buildings designed following seismic provisions of BNBC [1] would

behave quite conservatively during an earthquake. The major findings related to seismic

performance of building designed according to BNBC [1] can be summarized as follows:

1) Performance of building frames designed as per BNBC [1] has been evaluated

using pushover analysis, against targeted performance levels for serviceability,

design and maximum earthquakes and found to satisfy the ATC-40[3] local and

global requirements.

2) Performance evaluations of 2D RC frames were carried out using nonlinear time

history analysis and showed that frames designed by BNBC [1] for Dhaka city can

easily satisfy the local and global performance requirements.

3) Different RC buildings with varying story number, aspect ratio have been designed

as per BNBC [1]. Performances of these buildings have been carried out using 3D

pushover analysis. The building easily satisfies the seismic requirements of ATC-

40[3].

4) To demonstrate the importance of considering seismic load in design, a 3D building

has been designed without earthquake load and its performance has been checked

by pushover analysis. The building has been found to be inadequate for

serviceability and maximum earthquakes.

6.3 Recommendation for future study

This study employed a few number of reinforced concrete moment resisting frames and a

limited number of ground motion excitations. An extensive study containing a larger number

of frames and a set of representative ground motion records for Bangladesh would enhance

the results obtained in the accuracy of capacity of structure and seismic demand prediction of

pushover procedures.

This work may be extended in future to include the following:

1) More representative earthquake data should be considered for analyzing the

structures and to compare the resulting structural responses.

147

2) Three dimensional models of case study frames should be used for time history

analysis by considering the necessary geometric and strength characteristics of all

members that affect the nonlinear seismic response.

3) Experimental works should be carried out to verify the numerical results of 3D

frames.

4) Nonlinear time history analysis of RC 3D frames with soft story should be carried

out as it is an important issue in local residential areas.

5) Nonlinear behavior of RC structure with vertical irregularity should be studied to

assess their performance under a seismic event.

6) Different pushover analysis methods, i.e. adaptive pushover, modal pushover

should be studied in detail.

148

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1

Appendix A Table 2-A1 Modeling Parameters for Nonlinear Procedures – Reinforced Concrete Beams (ATC-40) Modeling Parameters3

Plastic Rotation Angle, rad

Residual Strength Ratio

Component Type a b c 1. Beam Controlled by Flexure1

balρρρ ′−

Transverse Reinforcement2

4

cw fdbV

′

≤0.0 C ≤3 0.025 0.05 0.2 ≤0.0 C ≥6 0.02 0.04 0.2 ≥0.5 C ≤3 0.02 0.03 0.2 ≥0.5 C ≥6 0.015 0.02 0.2 ≤0.0 NC ≤3 0.02 0.03 0.2 ≤0.0 NC ≥6 0.01 0.01

5 0.2

≥0.5 NC ≤3 0.01 0.015

0.2

≥0.5 NC ≥6 0.005 0.01 0.2 2. Beams controlled by shear1

Stirrup spacing ≤d/2 0.0 0.02 0.2 Stirrup spacing > d/2 0.0 0.01 0.2 3. Beams controlled by inadequate development or splicing along the span1

Stirrup spacing ≤d/2 0.0 0.02 0.0 Stirrup spacing >d/2 0.0 0.01 0.0 4. Beams controlled by inadequate embedment into beam-column joint1

0.015 0.03 0.2 1. When more than one of the conditions 1,2,3 and 4 occur for a given

component, use the minimum appropriate numerical value from the table. 2. Under the heading “transverse reinforcement,” ‘C’ and ‘NC’ are abbreviations

for conforming and non-conforming details, respectively. A component is conforming if within the flexural plastic region: (1) closed stirrup are spaced at ≤d/3 and 2) for components of moderate and high ductility demand the strength provided by the stirrup (Vs) is at least three-fourths of the design shear. Otherwise, the component is considered non-conforming.

3. Linear interpolation between values listed in the table is permitted

4. V = design shear force

2

Table 2-A2 Modeling Parameters for Nonlinear Procedures – Reinforced Concrete Column (ATC-40) Modeling Parameters4

Plastic Rotation Angle, rad

Residual Strength

Ratio Component Type a b c 1. Columns Controlled by Flexure1

5

cg fAP′

Transverse

Reinforcement2 6

cw fdbV

′

≤0.1 C ≤3 0.02 0.03 0.2 ≤0.1 C ≥6 0.015 0.025 0.2 ≥0.4 C ≤3 0.015 0.025 0.2 ≥0.4 C ≥6 0.01 0.015 0.2 ≤0.1 NC ≤3 0.01 0.015 0.2 ≤0.1 NC ≥6 0.005 0.005 - ≥0.4 NC ≤3 0.005 0.005 - ≥0.4 NC ≥6 0.0 0.0 -

2. Columns controlled by shear1

Hoop spacing ≤ d/2 or 5

cg fAP′

≤ 0.1 0.0 0.015 0.2

Other cases 0.0 0.0 0.0 3. Columns controlled by inadequate development or splicing along the clear height1,3

Hoop spacing ≤d/2 0.01 0.02 0.4 Hoop spacing >d/2 0.0 0.01 0.2 4. Column with axial loads exceeding 0.40 P0

1,3 Conforming reinforcement over the entire length

0.015 0.025 0.02

All other cases 0.0 0.0 0.0 1. When more than one of the conditions 1,2,3 and 4 occur for a given

component, use the minimum appropriate numerical value from the table.

2. Under the heading “transverse reinforcement,” ‘C’ and ‘NC’ are abbreviations for conforming and non-conforming details, respectively. A component is conforming if within the flexural plastic hinge region: (1) closed hoops are spaced at ≤d/3 and 2) for components of moderate and high ductility demand the strength provided by the stirrup (Vs) is at least three-fourths of the design shear. Otherwise, the component is considered non-conforming.

3. To quality, (1) hoops must not be lap spliced in the cover concrete, and (2) hoops must have hooks embedded in the core or must have other details to ensure that hoops will be adequately anchored following spalling of cover concrete.

4. Linear interpolation between values listed in the table is permitted.

5. P = Design axial load

6. V = design shear force

3

Table 2-A3 Modeling Parameters for Concrete Axial Hinge (FEMA-356, 2000) Modeling Parameters1

Plastic Deformation Residual Strength Ratio

Component Type a b c 1. Braces in Tension (except EBF braces)

11∆T 14∆T 0.8

1 ∆T is the axial deformation at expected tensile yielding load.

4

Table 2-A4 Modeling Parameters for Nonlinear Procedures-Coupling Beams (ATC-40)

Modeling Parameters3

Chord Rotation, rad

Residual Strength

Ratio Component Type d e c 1. Coupling beams controlled by flexureLongitudinal reinforcement and transverse reinforcement1

2

cw fdbV

′

Conventional longitudinal reinforcement with

≤3 0.025 0.040 0.75

Conforming transverse reinforcement

≥6 0.015 0.030 0.50

Conventional longitudinal reinforcement with non-

≤3 0.020 0.035 0.50


≥6 0.010 0.025 0.25

Diagonal reinforcement N/A 0.030 0.050 0.80 2. Coupling beams controlled by shear Longitudinal reinforcement and transverse reinforcement1

2

cw fdbV

′

Conventional longitudinal reinforcement with

≤3 0.018 0.030 0.60


≥6 0.012 0.020 0.30

Conventional longitudinal reinforcement with non-

≤3 0.012 0.025 0.40


≥6 0.008 0.014 0.20

1. Conventional longitudinal steel consists of top and bottom steel parallel to the

longitudinal axis of the beam. The requirements for conforming transverse reinforcement are: (1) closed stirrups are to be provided over the entire length of the beam at spacing not exceeding d/3; and (2) the strength provided by the stirrups (Vs) should be at least three-fourths of the design shear.

2. V = the design shear force on the coupling beam in pounds, bw = the web

width of the beam, d = the effective depth of the beam and fc’ = concrete compressive strength in psi.


5

Appendix B

Table 2-B1 Damage Control and Building Performance Levels (FEMA-356)

Target Building Performance Levels Collapse

Prevention Performance Level

Life Safety Performance Level

Immediate Occupancy

Performance Level

Operational Performance

Level Overall Damage

Severe Moderate Light Very Light

General Little residual stiffness and strength, but load-bearing columns and walls function. Large permanent drifts. Some exits blocked. Infills and unbraced parapets failed or at incipient failure. Building is near collapse

Some residual strength and stiffness left in all stories. Gravity-load-bearing elements function. No out-of-plane failure of walls or tipping of parapets. Some permanent drift. Damage to partitions. Building may be beyond economical repair.

No permanent drift. Structure substantially retains original strength and stiffness. Minor cracking of facades, partitions, and ceilings as well as structural elements. Elevators can be restarted. Fire protection operable.

No permanent drift. Structure substantially retains original strength and stiffness. Minor cracking of facades, partitions, and ceilings as well as structural elements. All systems important to normal operation are functional.

Nonstructural components

Extensive damage Falling hazards mitigated but many architectural, mechanical and electrical systems are damaged.

Equipment and contents are generally secure, but may not operate due to mechanical failure or lack of utilities.

Negligible damage occurs. Power and other utilities as available, possibly from standby sources.

Comparison with performance intended for buildings designed under the NEHRP Provisions, for the Design Earthquake

Significantly more damage and greater risk.

Somewhat more damage and slightly higher risk.

Less damage and lower risk.

Much less damage and lower risk.

6

Table 2-B2 Structural Performance Levels and Damage1.2.3 – Vertical Elements (FEMA-356)

Structural Performance Levels Collapse Prevention Life Safety Immediate

Occupancy Elements Type S-5 S-3 S-1 Concrete Frames

Primary Extensive cracking and hinge formation in ductile elements. Limited cracking and/or splice failure in some non-ductile columns. Severe damage in short columns

Extensive damage to beams. Spalling of cover and shear cracking (<1/8” width) for ductile columns. Minor spalling in non-ductile columns. Joint cracks <1/8” wide.

Minor hairline cracking. Limited yielding possible at a few locations. No crushing (strains below 0.003).

Secondary Extensive spalling in columns (limited shortening) and beams. Severe joint damage. Some reinforcing buckled

Extensive cracking and hinge formation in ductile elements. Limited cracking and/or splice failure in some nonductile columns. Severe damage in short columns

Minor spalling in non-ductile columns and beams. Flexural cracking in beams and columns. Shear cracking in Joint <1/6” width.

Drift 4% transient or permanent

2% transient; 1% permanent`

1% transient; negligible permanent

Steel Moment Frames

Primary Extensive distortion of beams and column panels. Many fractures at moment connections, but shear connections remain intact

Hinges form. Local bucking of some beam elements. Severe joint distortion; isolated moment connection fractures, but shear connections remain intact. A few elements may intact. A few elements may experience partial fracture.

Minor or local yielding at a few places. No fractures. Minor buckling or observable permanent distortion of members.

Secondary Same as primary Extensive distortion of beams and column panels. Many fractures at moment connections, but shear connections remain intact

Same as primary


2.5% transient; 1% permanent

0.7% transient; negligible permanent

Braced Steel Frames

Primary Extensive yielding and buckling of braces. Many braces and their connections may fail.

Many braces yield or buckle but do not totally fail. Many connections may fail

Minor yielding or buckling of braces.

7


Occupancy Elements Type S-5 S-3 S-1 Secondary Same as primary Same as primary Same as primary Drift 2% transient or

permanent 1.5% transient; 0.5% permanent


Concrete Walls

Primary Major flexural and shear cracks and voids. Sliding at joints. Extensive crushing and buckling of reinforcement. Failure around openings. Severe boundary element damage. Coupling beams shattered and virtually disintegrated.

Some boundary element stress, including limited buckling of reinforcement. Some sliding at joints. Damage around openings. Some crushing and flexural cracking. Coupling beams: extensive shear and flexural cracks; some crushing, but concrete generally remains in place.

Minor hairline cracking of walls, <1/16” wide. Coupling beams experience cracking <1/8” width.

Secondary Panels shattered and virtually disintegrated

Major flexural and shear cracks. Sliding at joints. Extensive crushing. failure around openings. Severe boundary element damage. Coupling beams shattered and virtually disintegrated.

Minor hairline cracking of walls. Some evidence of sliding at construction joints. Coupling beams experience cracks <1/8” width. Minor spalling.


1% transient; 0.5% permanent


Un-reinforced Masonry Infill Walls

Primary Extensive cracking and crushing; portions of face course shed

Extensive cracking and some crushing but wall remains in place. No falling units. Extensive crushing and spalling of veneers at corners of openings.

Minor (<1/8” width) cracking of masonry infill and veneers. Minor spalling in veneers at a few corner openings.

Secondary Extensive crushing and shattering; some walls dislodge.

Same as primary Same as primary

Drift 0.6% transient or permanent

0.5% transient; 0.3% permanent


Un-reinforced

Primary Extensive cracking; face course and veneer

Extensive cracking. Noticeable in-plane

Minor (<1/8” width) cracking of veneers.

8


Occupancy Elements Type S-5 S-3 S-1 Masonry (Non infill) Walls

may peel off. Noticeable in plane and out-of-plane offsets

offsets of masonry and minor out-of-plane offsets

Minor spalling in veneers at a few corner openings. No observable out-of-plane offsets.

Secondary Nonbearing panels dislodge

Same as primary Same as primary




Reinforced Masonry Walls

Primary Crushing; extensive cracking. Damage around openings and at corners. Some fallen units

Extensive cracking (<1/4”) distributed throughout wall. Some isolated crushing

Minor (<1/8” width) cracking. No out-of-plane offsets.

Secondary Panels shattered and virtually disintegrated

Crushing; extensive cracking. Damage around openings and at corners. Some fallen units

Same as primary

Drift 1.5% transient or permanent



Wood Stud Walls

Primary Connections loose. Nails partially withdrawn. Some splitting of members and panels. Veneers dislodged

Moderate loosening of connections and minor splitting of members

Distributed minor hairline cracking of gypsum and plaster veneers.

Secondary Sheathing sheared off. Let in braces fractured and buckled. Framing split and fractured

Connections loose. Nails partially withdrawn. Some splitting of members and panels.

Same as primary


2% transient; 1% permanent

1% transient;8 0.25% permanent

Precast Concrete Connections

Primary Some connection failures but no elements dislodged

Local crushing and spalling at connections, but no gross failure of connections.

Minor working at connections; cracks <1/16” width at connections.

Secondary Same as primary Some connection failures but no elements dislodged

Minor crushing and spalling at connections

Foundations

General Major settlement and tilting

Total settlements <6” and differential settlements <1/2” in 30ft.

Minor settlement and negligible tilting.

9

1. Damage states indicated in this table are provided to allow an understanding of the severity of damage that may be sustained by various structural elements when present in structures meeting the definitions of the Structural Performance Levels. These damage states are not intended for use in post-earthquake evaluation of damage or for judging the safety of, or required level of repair to, a structure following an earthquake.

2. Drift values, differential settlements, crack widths, and similar quantities

indicated in these tables are not intended to be used as acceptance criteria for evaluating the acceptability of a rehabilitation design in accordance with the analysis procedures provided in this standard; rather, they are indicative of the range of drift that typical structures containing the indicated structural elements may undergo when responding within the various Structural Performance Levels. Drift control of a rehabilitated structure may often be governed by the requirements to protect nonstructural components. Acceptable levels of foundation settlement or movement are highly dependent on the construction of the superstructure. The values indicated are intended to be qualitative descriptions of the approximate behavior of structures meeting the indicated levels.

3. For limiting damage to frame elements of infilled frames, refer to the rows for

concrete or steel frames.

10

Table 2-B3 Structural Performance Levels and Damage1.2 – Horizontal Elements (FEMA-356)


Occupancy Elements S-5 S-3 S-1 Metal Deck Diaphragms

Large distortion with buckling of some units and tearing of many welds and seam attachments

Some localized failure of welded connections of deck to framing and between panels. Minor local buckling of deck

Connections between deck units and framing intact. Minor distortions.

Wood Diaphragms

Large permanent distortion with partial withdrawal of nails and extensive splitting of elements

Some splitting at connections. Loosening of sheathing. Observable withdrawal of fasteners. Splitting of framing and sheathing.

No observable loosening or withdrawal of fasteners. No splitting of sheathing or framing.

Concrete Diaphragms

Extensive crushing and observable offset across many cracks.

Extensive cracking (<1/4” width). Local crushing and spalling

Distributed hairline cracking. Some minor cracks of larger size (<1/8” width).

Precast Diaphragms

Connections between units fail. Units shift relative to each other. Crushing and spalling at joints.

Extensive cracking (<1/4” width). Local crushing and spalling.

Some minor cracking along joints.

1. Damage states indicated in this table are provided to allow an understanding of

the severity of damage that may be sustained by various structural elements when present in structures meeting the definitions of the Structural Performance Levels. These damage states are not intended for use in post-earthquake evaluation of damage or for judging the safety of, or required level of repair to, a structure following an earthquake.

2. Drift values, differential settlements, crack widths, and similar quantities

indicated in these tables are not intended to be used as acceptance criteria for evaluating the acceptability of a rehabilitation design in accordance with the analysis procedures provided in this standard; rather, they are indicative of the range of drift that typical structures containing the indicated structural elements may undergo when responding within the various Structural Performance Levels. Drift control of a rehabilitated structure may often be governed by the requirements to protect nonstructural components. Acceptable levels of foundation settlement or movement are highly dependent on the construction of the superstructure. The values indicated are intended to be qualitative descriptions of the approximate behavior of structures meeting the indicated levels.

11

Appendix C Table 2-C1 Numerical Acceptance Criteria for Plastic Hinge Rotations in Reinforced Concrete Beams, in radians (ATC-40)

Performance Level3

Primary Secondary Component Type IO LS SS LS SS 1. Beams Controlled by Flexure1

balρρρ ′−

Transverse Reinforce

ment2

4

cw fdbV

′

≤0.0 C ≤3 0.005 0.020 0.025 0.020 0.050 ≤0.0 C ≥6 0.005 0.010 0.020 0.020 0.040 ≥0.5 C ≤3 0.005 0.010 0.020 0.020 0.030 ≥0.5 C ≥6 0.005 0.005 0.015 0.015 0.020 ≤0.0 NC ≤3 0.005 0.010 0.020 0.020 0.030 ≤0.0 NC ≥6 0.000 0.005 0.010 0.010 0.015 ≥0.5 NC ≤3 0.005 0.010 0.010 0.010 0.015 ≥0.5 NC ≥6 0.000 0.005 0.005 0.005 0.010

2. Beams controlled by shear1

Stirrup spacing ≤ d/2 0.0 0.0 0.0 0.010 0.020 Stirrup spacing > d/2 0.0 0.0 0.0 0.005 0.010 3. Beams controlled by inadequate development or splicing along the span1

Stirrup spacing ≤ d/2 0.0 0.0 0.0 0.010 0.020 Stirrup spacing > d/2 0.0 0.0 0.0 0.005 0.010 4. Beams controlled by inadequate embedment into beam-column joint1

0.0 0.01 0.015 0.020 0.030 1. When more than one of the conditions 1,2,3 and 4 occur for a given

component, use the minimum appropriate numerical value from the table. 2. Under the heading “transverse reinforcement,” ‘C’ and ‘NC’ are abbreviations

for conforming and non-conforming details, respectively. A component is conforming if within the flexural plastic region: (1) closed stirrup are spaced at ≤d/3 and 2) for components of moderate and high ductility demand the strength provided by the stirrup (Vs) is at least three-fourths of the design shear. Otherwise, the component is considered non-conforming.


IO = Immediate Occupancy LS = Life Safety SS = Structural Stability

4. V = Design Shear

12

Table 2-C2 Numerical Acceptance Criteria for Plastic Hinge Rotations in Reinforced Concrete Columns, in radians (ATC-40) Performance Level4

Primary Secondary Component Type IO LS SS LS SS 1. Columns Controlled by Flexure1

5

cg fAP′

Transverse

Reinforcement2

6

cw fdbV

′

≤0.1 C ≤3 0.005

0.010 0.020

0.015

0.030

≤0.1 C ≥6 0.005

0.010 0.015

0.010

0.025

≥0.4 C ≤3 0.000

0.005 0.015

0.010

0.025

≥0.4 C ≥6 0.000

0.005 0.010

0.010

0.015

≤0.1 NC ≤3 0.005

0.005 0.010

0.005

0.015

≤0.1 NC ≥6 0.005

0.005 0.005

0.005

0.005

≥0.4 NC ≤3 0.000

0.000 0.005

0.000

0.005

≥0.4 NC ≥6 0.000

0.000 0.000

0.000

0.000

2. Columns controlled by shear1,3

Hoop spacing ≤d/2,

or 1.0≤′cg fA

P

0.000

0.000 0.000

0.010

0.015

Other cases 0.000

0.000 0.000

0.00 0.000

3. Columns controlled by inadequate development or splicing along the clear height1,3

Hoop spacing ≤d/2 0.0 0.0 0.0 0.010

0.020

Hoop spacing >d/2 0.0 0.0 0.0 0.005

0.010

4. Columns with axial loads exceeding 0.70 1,3

Conforming reinforcement over the entire length

0.0 0.0 0.005

0.005

0.010

All other cases 0.0 0.0 0.0 0.0 0.0 1. When more than one of the conditions 1, 2, 3 and 4 occur for a given

component, use the minimum appropriate numerical value from the table. See Chapter 9 for symbol definitions.

2. Under the heading “transverse reinforcement,” ‘C’ and ‘NC’ are abbreviations for conforming and non-conforming details, respectively. A component is conforming if within the flexural plastic hinge region: (1) closed hoops are spaced at ≤d/3 and 2) for components of moderate and high ductility demand

13

the strength provided by the stirrup (Vs) is at least three-fourths of the design shear. Otherwise, the component is considered non-conforming.

3. To qualify, (1) hoops must not be lap spliced in the cover concrete, and (2) hoops must have hooks embedded in the core or must have other details to ensure that hoops will be adequately anchored following spalling of cover concrete.

4. Linear interpolation between values listed in the table is permitted. IO = Immediate Occupancy LS = Life Safety SS = Structural Stability

5. P = Design axial load 6. V = Design shear force

14

Table 2-C3 Numerical Acceptance Criteria for Chord Rotations for Reinforced Concrete Coupling Beams

Performance Level3

Primary Secondary Component Type IO LS SS LS SS 1. Coupling beams controlled by flexureLongitudinal reinforcement and transverse reinforcement1

2

cw fdbV

′

Conventional longitudinal reinforcement with conforming transverse reinforcement

≤3 0.006 0.015 0.025 0.025 0.040


≥6 0.005 0.010 0.015 0.015 0.030

Conventional longitudinal reinforcement with non-conforming transverse reinforcement

≤3 0.006 0.012 0.020 0.020 0.035

Conventional longitudinal reinforcement with non-conforming transverse reinforcement

≥6 0.005 0.008 0.010 0.010 0.025

Diagonal reinforcement N/A 0.006 0.018 0.030 0.030 0.050 2. Coupling beams controlled by shear Longitudinal reinforcement and transverse reinforcement1

2

cw fdbV

′


≤3 0.006 0.012 0.015 0.015 0.024


≥6 0.004 0.008 0.010 0.010 0.016

Conventional longitudinal reinforcement with non- conforming transverse reinforcement

≤3 0.006 0.008 0.010 0.010 0.020

Conventional longitudinal reinforcement with non- conforming transverse reinforcement

≥6 0.004 0.006 0.007 0.007 0.012

1. Conventional longitudinal steel consists of top and bottom steel parallel to the

longitudinal axis of the beam. The requirements for conforming transverse reinforcement are: (1) closed stirrups are to be provided over the entire length of the beam at spacing not exceeding d/3; and (2) the strength provided by the stirrups (Vs) should be at least three-fourths of the design shear.

2. V = the design shear force on the coupling beam in pounds, bw = the web

width of the beam, d = the effective depth of the beam and f’c = concrete compressive strength in psi.


IO = Immediate occupancy LS = Life Safety SS = Structural Stability

15

Table 2-C4 Numerical Acceptance Criteria for Reinforced Concrete Column Axial Hinge (FEMA-356)

Plastic Deformation1

Primary Secondary Component Type IO LS SS LS SS 1. Braces in Tension (except EBF braces)

7∆T 9∆T 11∆T 11∆T 13∆T

1 ∆T is the axial deformation at expected tensile yielding load.

16

Table 2-C5 Numerical Acceptance Criteria for Total Shear Angle in Reinforced Concrete Beam-Columns Joints, in radians (ATC-40, 1996)

Performance Level4

Primary6 Secondary Component Type IO LS SS LS SS 1. Interior joints

2

cg fAP′

Transverse

Reinforcement1 3

nVV

≤0.1 C ≤1.2 0.0 0.0 0.0 0.020 0.030 ≤0.1 C ≥1.5 0.0 0.0 0.0 0.015 0.020 ≥0.4 C ≤1.2 0.0 0.0 0.0 0.015 0.025 ≥0.4 C ≥1.5 0.0 0.0 0.0 0.015 0.020 ≤0.1 NC ≤1.2 0.0 0.0 0.0 0.015 0.020 ≤0.1 NC ≥1.5 0.0 0.0 0.0 0.010 0.015 ≥0.4 NC ≤1.2 0.0 0.0 0.0 0.010 0.015 ≥0.4 NC ≥1.5 0.0 0.0 0.0 0.010 0.015

2. Other joints2

cg fAP′

Transverse

Reinforcement1 3

nVV

≤0.1 C ≤1.2 0.0 0.0 0.0 0.015 0.020 ≤0.1 C ≥1.5 0.0 0.0 0.0 0.010 0.015 ≥0.4 C ≤1.2 0.0 0.0 0.0 0.015 0.020 ≥0.4 C ≥1.5 0.0 0.0 0.0 0.010 0.015 ≤0.1 NC ≤1.2 0.0 0.0 0.0 0.005 0.010 ≤0.1 NC ≥1.5 0.0 0.0 0.0 0.005 0.010 ≥0.4 NC ≤1.2 0.0 0.0 0.0 0.000 0.000 ≥0.4 NC ≥1.5 0.0 0.0 0.0 0.000 0.000

1. Under the heading “transverse reinforcement,” ‘C’ and ‘NC’ are abbreviations for conforming and non-conforming details, respectively. A joint is conforming if closed hoops are spaced at ≤hc/3 within the joint. Otherwise, the component is considered non-conforming. Also, to qualify as conforming details under condition 2, (1) closed hoops must not be lap spliced in the cover concrete, and (2) hoops must have hooks embedded in the core or must have other details to ensure that hoops will be adequately anchored following spalling of cover concrete.

2. The ratio cg fA

P′

is the ratio of the design axial force on the column above the

joint to the product of the gross cross-sectional and lateral forces. 3. The ratio V/Vn is the ratio of the design shear force to the shear strength for

the joint. 4. Linear interpolation between values listed in the table is permitted.


5. No inelastic deformation is permitted since joint yielding is not allowed in a conforming building.

17

Table 2-C6 Numerical Acceptance Criteria for Plastic Hinge Rotation in Reinforced Concrete Two-way Slabs and Slab-Column Connections, in radians (ATC-40)

Performance Level4

Primary Secondary Component Type IO LS SS LS SS 1. Slabs controlled by flexure and slab column connections1

2

0VVg

Continuity Reinforcement3

≤0.2 Yes 0.01 0.015 0.02 0.030 0.05 ≥0.4 Yes 0.00 0.000 0.00 0.030 0.04 ≤0.2 No 0.01 0.015 0.02 0.015 0.02 ≥0.4 No 0.00 0.000 0.00 0.000 0.00

2. Slabs controlled by inadequate development or splicing along the span1 0.00 0.00 0.000 0.01 0.02

3. Slabs controlled by inadequate embedment into slab-column joint1

0.01 0.01 0.015 0.02 0.03 1. When more than one of the conditions 1, 2, 3 and 4 occur for a given

component, use the minimum appropriate numerical value from the table. 2. Vg = the gravity shear acting on the slab critical section as defined by ACI

318, Vo = the direct punching shear strength as defined by ACI 318. 3. Under the heading “Continuity reinforcement” assume ‘Yes’ where at least

one of the main bottom bars in each direction is effectively continuous through the column cage. Where the slab is post-tensioned, assume “Yes” where at least one of the post-tensioning tendons in each direction passes through the column cage. Otherwise, assume “No.”



18

Table 2-C7 Numerical Acceptance Criteria for Plastic Hinge Rotations in Reinforced Concrete Walls and Wall Segments Controlled by Flexure, in radians (ATC-40)

Performance Level4

Primary Secondary Component Type IO LS SS LS SS 1. Walls and wall segments controlled by flexure

1)(′+′−

cww

ySS

flt

PfAA

2

′cww flt

V

Boundary Element3

≤0.1 ≤3 C 0.005 0.010 0.015 0.015 0.020 ≤0.1 ≥6 C 0.004 0.008 0.010 0.010 0.015 ≥0.25 ≤3 C 0.003 0.006 0.009 0.009 0.012 ≥0.25 ≥6 C 0.001 0.003 0.005 0.005 0.010 ≤0.1 ≤3 NC 0.002 0.004 0.008 0.008 0.015 ≤0.1 ≥6 NC 0.002 0.004 0.006 0.006 0.010 ≥0.25 ≤3 NC 0.001 0.002 0.003 0.003 0.005 ≥0.25 ≥6 NC 0.001 0.001 0.002 0.002 0.004

1. As = the cross-sectional area of longitudinal reinforcement in tension, As’ = the

cross-sectional area of longitudinal reinforcement in compression, fy = yield stress of longitudinal reinforcement, P = axial force acting on the wall considering design load combinations, tw = wall web thickness, lw = wall length, and f’c = concrete compressive strength.

2. V = the design shear force acting on the wall, and other variables are as

defined above. 3. The term “C” indicates the boundary reinforcement effectively satisfies

requirements of ACI 318. The term “NC” indicates the boundary requirements do not satisfy requirements of ACI 318.

4. Linear interpolation between values listed in the table is permitted. IO = Immediate Occupancy

LS = Life Safety

SS = Structural Stability

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